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1,964 | EP-158 | Erdős Problem #158 | Let $A\subset \mathbb{N}$ be an infinite set such that, for any $n$, there are most $2$ solutions to $a+b=n$ with $a\leq b$. Must $ \liminf_{N\to\infty}\frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/2}}=0? $ | If we replace $2$ by $1$ then $A$ is a Sidon set, for which Erd\H{o}s proved this is true.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** Whether every infinite B2[2] set has... | 1 | open | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,965 | EP-159 | Erdős Problem #159 | There exists some constant $c>0$ such that
$$R(C_4,K_n) \ll n^{2-c}.$$ | The current bounds are $ \frac{n^{3/2}}{(\log n)^{3/2}}\ll R(C_4,K_n)\ll \frac{n^2}{(\log n)^2}. $ The upper bound is due to Szemer\'{e}di (mentioned in \cite{EFRS78}), and the lower bound is due to Spencer \cite{Sp77}.
This problem is #17 in Ramsey Theory in the graphs problem collection.
References
[EFRS78] Erd\H{... | 1 | open | null | null | 3 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,966 | EP-160 | Erdős Problem #160 | Let $h(N)$ be the smallest $k$ such that $\{1,\ldots,N\}$ can be coloured with $k$ colours so that every four-term arithmetic progression must contain at least three distinct colours. Estimate $h(N)$. | Investigated by Erd\H{o}s and Freud. This has been discussed on MathOverflow, where LeechLattice shows $ h(N) \ll N^{2/3}. $ In the comments of this site Hunter improves this to $ h(N) \ll N^{\frac{\log 3}{\log 22}+o(1)} $ (note $\frac{\log 3}{\log 22}\approx 0.355$).
The observation of Zach Hunter in that question cou... | 1 | open | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,967 | EP-161 | Erdős Problem #161 | Let $\alpha\in[0,1/2)$ and $n,t\geq 1$. Let $F^{(t)}(n,\alpha)$ be the smallest $m$ such that we can $2$-colour the edges of the complete $t$-uniform hypergraph on $n$ vertices such that if $X\subseteq [n]$ with $\lvert X\rvert \geq m$ then there are at least $\alpha \binom{\lvert X\rvert}{t}$ many $t$-subsets of $X$ o... | For $\alpha=0$ this is the usual Ramsey function.
A conjecture of Erd\H{o}s, Hajnal, and Rado (see [562]) implies that $ F^{(t)}(n,0)\asymp \log_{t-1} n $ and results of Erd\H{o}s and Spencer imply that $ F^{(t)}(n,\alpha) \gg_\alpha (\log n)^{\frac{1}{t-1}} $ for all $\alpha>0$, and a similar upper bound holds for $\... | 1 | partially_solved | null | null | 3 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,968 | EP-162 | Erdős Problem #162 | Let $\alpha>0$ and $n\geq 1$. Let $F(n,\alpha)$ be the largest $k$ such that there exists some 2-colouring of the edges of $K_n$ in which any induced subgraph $H$ on at least $k$ vertices contains more than $\alpha\binom{\lvert H\rvert}{2}$ many edges of each colour.
Prove that for every fixed $0\leq \alpha \leq 1/2$, ... | It is easy to show with the probabilistic method that there exist $c_1(\alpha),c_2(\alpha)$ such that $ c_1(\alpha)\log n < F(n,\alpha) < c_2(\alpha)\log n. $ ",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PAR... | 1 | partially_solved | null | null | 3 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,969 | EP-165 | Erdős Problem #165 | Give an asymptotic formula for $R(3,k)$. | It is known that there exists some constant $c>0$ such that for large $k$ $ (c+o(1))\frac{k^2}{\log k}\leq R(3,k) \leq (1+o(1))\frac{k^2}{\log k}. $ The lower bound is due to Kim \cite{Ki95}, the upper bound is due to Shearer \cite{Sh83}, improving an earlier bound of Ajtai, Koml\'{o}s, and Szemer\'{e}di \cite{AKS80}.
... | 1 | open | null | null | 3 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,970 | EP-168 | Erdős Problem #168 | Let $F(N)$ be the size of the largest subset of $\{1,\ldots,N\}$ which does not contain any set of the form $\{n,2n,3n\}$. What is $ \lim_{N\to \infty}\frac{F(N)}{N}? $ Is this limit irrational? | This limit was proved to exist by Graham, Spencer, and Witsenhausen \cite{GSW77}, who showed it is equal to $ \frac{1}{3}\sum_{k\in K}\frac{1}{d_k}, $ where $d_1<d_2<\cdots $are the $3$-smooth numbers and $K$ is the set of $k$ for which $f(k)>f(k-1)$, where $f$ counts the largest subset of $\{d_1,\ldots,d_k\}$ that avo... | 1 | partially_solved | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,971 | EP-169 | Erdős Problem #169 | Let $k\geq 3$ and $f(k)$ be the supremum of $\sum_{n\in A}\frac{1}{n}$ as $A$ ranges over all sets of positive integers which do not contain a $k$-term arithmetic progression. Estimate $f(k)$.
Is $ \lim_{k\to \infty}\frac{f(k)}{\log W(k)}=\infty $ where $W(k)$ is the van der Waerden number? | Berlekamp \cite{Be68} proved $f(k) \geq \frac{\log 2}{2}k$. Gerver \cite{Ge77} proved $ f(k) \geq (1-o(1))k\log k. $ It is trivial that $ \frac{f(k)}{\log W(k)}\geq \frac{1}{2}, $ but improving the right-hand side to any constant $>1/2$ is open.
Gerver also proved (see the comments for an alternative argument of Tao) t... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,972 | EP-170 | Erdős Problem #170 | Let $F(N)$ be the smallest possible size of $A\subset \{0,1,\ldots,N\}$ such that $\{0,1,\ldots,N\}\subset A-A$. Find the value of $ \lim_{N\to \infty}\frac{F(N)}{N^{1/2}}. $ | The Sparse Ruler problem. R\'{e}dei asked whether this limit exists, which was proved by Erd\H{o}s and G\'{a}l \cite{ErGa48}. Bounds on the limit were improved by Leech \cite{Le56}. The limit is known to be in the interval $[1.56,\sqrt{3}]$. The lower bound is due to Leech \cite{Le56}, the upper bound is due to Wichman... | 1 | open | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,973 | EP-172 | Erdős Problem #172 | Is it true that in any finite colouring of $\mathbb{N}$ there exist arbitrarily large finite $A$ such that all sums and products of distinct elements in $A$ are the same colour? | First asked by Hindman. Hindman \cite{Hi80} has proved this is false (with 7 colours) if we ask for an infinite $A$. In \cite{Er77c} Erd\H{o}s asks about the case for an infinite $A$ with just $2$ colours.
Moreira \cite{Mo17} has proved that in any finite colouring of $\mathbb{N}$ there exist $x,y$ such that $\{x,x+y,x... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,974 | EP-173 | Erdős Problem #173 | In any $2$-colouring of $\mathbb{R}^2$, for all but at most one triangle $T$, there is a monochromatic congruent copy of $T$. | For some colourings a single equilateral triangle has to be excluded, considering the colouring by alternating strips. Shader \cite{Sh76} has proved this is true if we just consider a single right-angled triangle.
References
[Sh76] Shader, L., All right triangles are Ramsey in $\mathbbE^2$!. J. Comb. Th. A (1976), 38... | 1 | open | null | null | 3 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,975 | EP-174 | Erdős Problem #174 | A finite set $A\subset \mathbb{R}^n$ is called Ramsey if, for any $k\geq 1$, there exists some $d=d(A,k)$ such that in any $k$-colouring of $\mathbb{R}^d$ there exists a monochromatic copy of $A$. Characterise the Ramsey sets in $\mathbb{R}^n$. | Erd\H{o}s, Graham, Montgomery, Rothschild, Spencer, and Straus \cite{EGMRSS73} proved that every Ramsey set is 'spherical': it lies on the surface of some sphere. Graham has conjectured that every spherical set is Ramsey. Leader, Russell, and Walters \cite{LRW12} have alternatively conjectured that a set is Ramsey if a... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,976 | EP-176 | Erdős Problem #176 | Let $N(k,\ell)$ be the minimal $N$ such that for any $f:\{1,\ldots,N\}\to\{-1,1\}$ there must exist a $k$-term arithmetic progression $P$ such that $ \left\lvert \sum_{n\in P}f(n)\right\rvert\geq \ell. $ Find good upper bounds for $N(k,\ell)$. Is it true that for any $c>0$ there exists some $C>1$ such that $ N(k,ck)\l... | When $\ell=k$ this is the van der Waerden number $W(k)$ (see [138]). Spencer \cite{Sp73} has proved that if $k=2^tm$ with $m$ odd then $ N(k,1)=2^t(k-1)+1. $ Erd\H{o}s and Graham write that 'no decent bound' is known even for $N(k,2)$.
Erd\H{o}s \cite{Er63d} proved that, for every $c>0$, $ N(k,ck)> (1+\alpha_c)^k $ whe... | 1 | open | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
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"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,977 | EP-177 | Erdős Problem #177 | Find the smallest $h(d)$ such that the following holds. There exists a function $f:\mathbb{N}\to\{-1,1\}$ such that, for every $d\geq 1$, $ \max_{P_d}\left\lvert \sum_{n\in P_d}f(n)\right\rvert\leq h(d), $ where $P_d$ ranges over all finite arithmetic progressions with common difference $d$. | Cantor, Erd\H{o}s, Schreiber, and Straus \cite{Er66} proved that $h(d)\ll d!$ is possible. Van der Waerden's theorem implies that $h(d)\to \infty$. Beck \cite{Be17} has shown that $h(d) \leq d^{8+\epsilon}$ is possible for every $\epsilon>0$. Roth's famous discrepancy lower bound \cite{Ro64} implies that $h(d)\gg d^{1/... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,978 | EP-180 | Erdős Problem #180 | If $\mathcal{F}$ is a finite set of finite graphs then $\mathrm{ex}(n;\mathcal{F})$ is the maximum number of edges a graph on $n$ vertices can have without containing any subgraphs from $\mathcal{F}$. Note that it is trivial that $\mathrm{ex}(n;\mathcal{F})\leq \mathrm{ex}(n;G)$ for every $G\in\mathcal{F}$.
Is it true ... | A problem of Erd\H{o}s and Simonovits.
This is trivially true if $\mathcal{F}$ does not contain any bipartite graphs, since by the Erd\H{o}s-Stone theorem if $H\in\mathcal{F}$ has minimal chromatic number $r\geq 2$ then $ \mathrm{ex}(n;H)=\mathrm{ex}(n;\mathcal{F})=\left(\frac{r-2}{r-1}+o(1)\right)\binom{n}{2}. $ Erd\H... | 1 | solved | null | null | 3 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | The maintained Erdős Problems source lists problem 180 as “disproved (Lean)” on 2026-09-27. The source marks a Lean verification of the resolution.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://www.erdosproblems.com/180",
"label": "Erdős Problems #180: disproved (Lean)"
},
{
"url": "https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml",
"label": "Erdős Problems machine-readable status list (snapshot checked 2026-09-27)"
},
{
"url": "https://hug... | 2026-09-27T00:00:00 | Erdős Problems #180: disproved (Lean): https://www.erdosproblems.com/180
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27T00:00:00",
"kind": "resolution",
"evidence": "maintained_source"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,979 | EP-181 | Erdős Problem #181 | Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges). Prove that $ R(Q_n) \ll 2^n. $ | Conjectured by Burr and Erd\H{o}s, althouhg in \cite{Er93} Erd\H{o}s says the behaviour of $R(Q_n)$ was considered by himself and S\'{o}s, who could not decide whether $R(Q_n)/2^n\to \infty$ or not.
The trivial bound is $ R(Q_n) \leq R(K_{2^n})\leq C^{2^n} $ for some constant $C>1$. This was improved a number of times;... | 1 | open | null | null | 3 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,980 | EP-183 | Erdős Problem #183 | Let $R(3;k)$ be the minimal $n$ such that if the edges of $K_n$ are coloured with $k$ colours then there must exist a monochromatic triangle. Determine $ \lim_{k\to \infty}R(3;k)^{1/k}. $ | Erd\H{o}s offers \$100 for showing that this limit is finite. An easy pigeonhole argument shows that $ R(3;k)\leq 2+k(R(3;k-1)-1), $ from which $R(3;k)\leq \lceil e k!\rceil$ immediately follows. The best-known upper bounds are all of the form $ck!+O(1)$, and arise from this type of inductive relationship and computati... | 1 | solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | The maintained Erdős Problems source lists problem 183 as “solved (Lean)” on 2026-09-27. The source marks a Lean verification of the resolution.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://www.erdosproblems.com/183",
"label": "Erdős Problems #183: solved (Lean)"
},
{
"url": "https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml",
"label": "Erdős Problems machine-readable status list (snapshot checked 2026-09-27)"
},
{
"url": "https://huggin... | 2026-09-27T00:00:00 | Erdős Problems #183: solved (Lean): https://www.erdosproblems.com/183
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27T00:00:00",
"kind": "resolution",
"evidence": "maintained_source"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,981 | EP-184 | Erdős Problem #184 | Any graph on $n$ vertices can be decomposed into $O(n)$ many edge-disjoint cycles and edges. | Conjectured by Erd\H{o}s and Gallai, who proved that $O(n\log n)$ many cycles and edges suffices. The graph $K_{3,n-3}$ shows that at least $(1+c)n$ many cycles and edges are required, for some constant $c>0$. In \cite{Er71} Erd\H{o}s suggests that only $n-1$ many cycles and edges are required if we do not require them... | 1 | open | null | null | 3 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,982 | EP-187 | Erdős Problem #187 | Find the best function $f(d)$ such that, in any 2-colouring of the integers, at least one colour class contains an arithmetic progression with common difference $d$ of length $f(d)$ for infinitely many $d$. | Originally asked by Cohen. Erd\H{o}s observed that colouring according to whether $\{ \sqrt{2}n\}<1/2$ or not implies $f(d) \ll d$ (using the fact that $\|\sqrt{2}q\| \gg 1/q$ for all $q$, where $\|x\|$ is the distance to the nearest integer). Beck \cite{Be80} has improved this using the probabilistic method, construct... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,983 | EP-188 | Erdős Problem #188 | What is the smallest $k$ such that $\mathbb{R}^2$ can be red/blue coloured with no pair of red points unit distance apart, and no $k$-term arithmetic progression of blue points with distance $1$? | Erd\H{o}s, Graham, Montgomery, Rothschild, Spencer, and Straus \cite{EGMRSS75} proved $k\geq 5$. Tsaturian \cite{Ts17} improved this to $k\geq 6$. Erd\H{o}s and Graham claim that $k\leq 10000000$ ('more or less'), but give no proof.
Erd\H{o}s and Graham asked this with just any $k$-term arithmetic progression in blue (... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,984 | EP-190 | Erdős Problem #190 | Let $H(k)$ be the smallest $N$ such that in any finite colouring of $\{1,\ldots,N\}$ (into any number of colours) there is always either a monochromatic $k$-term arithmetic progression or a rainbow arithmetic progression (i.e. all elements are different colours). Estimate $H(k)$. Is it true that $ H(k)^{1/k}/k \to \inf... | This type of problem belongs to 'canonical' Ramsey theory. The existence of $H(k)$ follows from Szemer\'{e}di's theorem, and it is easy to show that $H(k)^{1/k}\to\infty$.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-09-27)
**Status:** solved
**Classification:** SOL... | 1 | solved | null | null | 3 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | The maintained Erdős Problems source lists problem 190 as “solved (Lean)” on 2026-09-27. The source marks a Lean verification of the resolution.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://www.erdosproblems.com/190",
"label": "Erdős Problems #190: solved (Lean)"
},
{
"url": "https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml",
"label": "Erdős Problems machine-readable status list (snapshot checked 2026-09-27)"
},
{
"url": "https://huggin... | 2026-09-27T00:00:00 | Erdős Problems #190: solved (Lean): https://www.erdosproblems.com/190
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27T00:00:00",
"kind": "resolution",
"evidence": "maintained_source"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,985 | EP-193 | Erdős Problem #193 | Let $S\subseteq \mathbb{Z}^3$ be a finite set and let $A=\{a_1,a_2,\ldots,\}\subset \mathbb{Z}^3$ be an infinite $S$-walk, so that $a_{i+1}-a_i\in S$ for all $i$. Must $A$ contain three collinear points? | Originally conjectured by Gerver and Ramsey \cite{GeRa79}, who showed that the answer is yes for $\mathbb{Z}^2$, and for $\mathbb{Z}^3$ that the largest number of collinear points can be bounded.
References
[GeRa79] Gerver, Joseph L. and Ramsey, L. Thomas, On certain sequences of lattice points. Pacific J. Math. (197... | 1 | solved | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | The maintained Erdős Problems source lists problem 193 as “disproved (Lean)” on 2026-09-27. The source marks a Lean verification of the resolution.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://www.erdosproblems.com/193",
"label": "Erdős Problems #193: disproved (Lean)"
},
{
"url": "https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml",
"label": "Erdős Problems machine-readable status list (snapshot checked 2026-09-27)"
},
{
"url": "https://hug... | 2026-09-27T00:00:00 | Erdős Problems #193: disproved (Lean): https://www.erdosproblems.com/193
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27T00:00:00",
"kind": "resolution",
"evidence": "maintained_source"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,986 | EP-195 | Erdős Problem #195 | What is the largest $k$ such that in any permutation of $\mathbb{Z}$ there must exist a monotone $k$-term arithmetic progression $x_1<\cdots<x_k$? | Geneson \cite{Ge19} proved that $k\leq 5$. Adenwalla \cite{Ad22} proved that $k\leq 4$.
See also [194] and [196].
References
[Ad22] Adenwalla, S., Avoiding Monotone Arithmetic Progressions in Permutations of Integers. arXiv:2211.04451 (2022).
[Ge19] Geneson, Jesse, Forbidden arithmetic progressions in permutations o... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,987 | EP-196 | Erdős Problem #196 | Must every permutation of $\mathbb{N}$ contain a monotone 4-term arithmetic progression? In other words, given a permutation $x$ of $\mathbb{N}$ must there be indices with either $i<j<k<l$ or $i>j>k>l$ such that $x_i,x_j,x_k,x_l$ are an arithmetic progression? | Davis, Entringer, Graham, and Simmons \cite{DEGS77} have shown that there must exist a monotone 3-term arithmetic progression and need not contain a 5-term arithmetic progression.
See also [194] and [195].
References
[DEGS77] Davis, J. A. and Entringer, R. C. and Graham, R. L. and
Simmons, G. J., On permutations cont... | 1 | open | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,988 | EP-197 | Erdős Problem #197 | Can $\mathbb{N}$ be partitioned into two sets, each of which can be permuted to avoid monotone 3-term arithmetic progressions? | If three sets are allowed then this is possible.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** It remains open whether N can be partitioned into two subsets that each admit ... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,989 | EP-200 | Erdős Problem #200 | Does the longest arithmetic progression of primes in $\{1,\ldots,N\}$ have length $o(\log N)$? | It follows from the prime number theorem that such a progression has length $\leq(1+o(1))\log N$.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** The requested o(log N) bound ... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,990 | EP-201 | Erdős Problem #201 | Let $G_k(N)$ be such that any set of $N$ integers contains a subset of size at least $G_k(N)$ which does not contain a $k$-term arithmetic progression. Determine the size of $G_k(N)$. How does it relate to $R_k(N)$, the size of the largest subset of $\{1,\ldots,N\}$ without a $k$-term arithmetic progression? Is it true... | First asked and investigated by Riddell \cite{Ri69}. It is trivial that $G_k(N)\leq R_k(N)$, and it is possible that $G_k(N) <R_k(N)$ (for example $G_3(5)=3$ and $R_3(5)=4$, and $G_3(14)\leq 7$ and $R_3(14)=8$).
Koml\'{o}s, Sulyok, and Szemer\'{e}di \cite{KSS75} have shown that $R_k(N) \ll_k G_k(N)$.
References
[KSS7... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
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} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,991 | EP-202 | Erdős Problem #202 | Let $n_1<\cdots < n_r\leq N$ with associated $a_i\pmod{n_i}$ such that the congruence classes are disjoint (that is, every integer is $\equiv a_i\pmod{n_i}$ for at most one $1\leq i\leq r$). How large can $r$ be in terms of $N$? | Let $f(N)$ be the maximum possible $r$. Erd\H{o}s and Stein conjectured that $f(N)=o(N)$, which was proved by Erd\H{o}s and Szemer\'{e}di \cite{ErSz68}, who showed that, for every $\epsilon>0$, $ \frac{N}{\exp((\log N)^{1/2+\epsilon})} \ll_\epsilon f(N) < \frac{N}{(\log N)^c} $ for some $c>0$. Erd\H{o}s believed the lo... | 1 | solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 1,
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"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | The maintained Erdős Problems source lists problem 202 as “solved (Lean)” on 2026-09-27. The source marks a Lean verification of the resolution.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://www.erdosproblems.com/202",
"label": "Erdős Problems #202: solved (Lean)"
},
{
"url": "https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml",
"label": "Erdős Problems machine-readable status list (snapshot checked 2026-09-27)"
},
{
"url": "https://huggin... | 2026-09-27T00:00:00 | Erdős Problems #202: solved (Lean): https://www.erdosproblems.com/202
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27T00:00:00",
"kind": "resolution",
"evidence": "maintained_source"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,992 | EP-203 | Erdős Problem #203 | Is there an integer $m\geq 1$ with $(m,6)=1$ such that none of $2^k3^\ell m+1$ are prime, for any $k,\ell\geq 0$? | Positive odd integers $m$ such that none of $2^km+1$ are prime are called Sierpinski numbers - see [1113] for more details.
Erd\H{o}s and Graham also ask more generally about $p_1^{k_1}\cdots p_r^{k_r}m+1$ for distinct primes $p_i$, or $q_1\cdots q_rm+1$ where the $q_i$ are primes congruent to $1\pmod{4}$. (Dogmachine ... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
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} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
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"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,993 | EP-208 | Erdős Problem #208 | Let $s_1<s_2<\cdots$ be the sequence of squarefree numbers. Is it true that, for any $\epsilon>0$ and large $n$, $ s_{n+1}-s_n \ll_\epsilon s_n^{\epsilon}? $ Is it true that $ s_{n+1}-s_n \leq (1+o(1))\frac{\pi^2}{6}\frac{\log s_n}{\log\log s_n}? $ | Erd\H{o}s \cite{Er51} showed that there are infinitely many $n$ such that $ s_{n+1}-s_n > (1+o(1))\frac{\pi^2}{6}\frac{\log s_n}{\log\log s_n}, $ so this bound would be the best possible.
In \cite{Er79} Erd\H{o}s says perhaps $s_{n+1}-s_n \ll \log s_n$, but he is 'very doubtful'.
Filaseta and Trifonov \cite{FiTr92} pro... | 1 | open | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
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} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,994 | EP-212 | Erdős Problem #212 | Is there a dense subset of $\mathbb{R}^2$ such that all pairwise distances are rational? | Conjectured by Ulam. Erd\H{o}s believed there cannot be such a set. This problem is discussed in a blogpost by Terence Tao, in which he shows that there cannot be such a set, assuming the Bombieri-Lang conjecture. The same conclusion was independently obtained by Shaffaf \cite{Sh18}.
Indeed, Shaffaf and Tao actually pr... | 1 | open | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
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"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
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"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
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"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,995 | EP-213 | Erdős Problem #213 | Let $n\geq 4$. Are there $n$ points in $\mathbb{R}^2$, no three on a line and no four on a circle, such that all pairwise distances are integers? | Anning and Erd\H{o}s \cite{AnEr45} proved there cannot exist an infinite such set. Harborth constructed such a set when $n=5$. The best construction to date, due to Kreisel and Kurz \cite{KK08}, has $n=7$.
Ascher, Braune, and Turchet \cite{ABT20} have shown that there is a uniform upper bound on the size of such a set,... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
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"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,996 | EP-217 | Erdős Problem #217 | For which $n$ are there $n$ points in $\mathbb{R}^2$, no three on a line and no four on a circle, which determine $n-1$ distinct distances and so that (in some ordering of the distances) the $i$th distance occurs $i$ times? | An example with $n=4$ is an isosceles triangle with the point in the centre. Erd\H{o}s originally believed this was impossible for $n\geq 5$, but Pomerance constructed a set with $n=5$ (see \cite{Er83c} for a description), and Pal\'{a}sti has proved such sets exist for all $n\leq 8$.
Erd\H{o}s believed this is impossib... | 1 | open | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
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"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,997 | EP-218 | Erdős Problem #218 | Let $d_n=p_{n+1}-p_n$. The set of $n$ such that $d_{n+1}\geq d_n$ has density $1/2$, and similarly for $d_{n+1}\leq d_n$. Furthermore, there are infinitely many $n$ such that $d_{n+1}=d_n$. | In \cite{Er85c} Erd\H{o}s also conjectures that $d_n=d_{n+1}=\cdots=d_{n+k}$ is solvable for every $k$ (which is equivalent to $k$ consecutive primes in arithmetic progression, see [141]).
References
[Er85c] Erd\H{o}s, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacam... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
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} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,998 | EP-222 | Erdős Problem #222 | Let $n_1<n_2<\cdots$ be the sequence of integers which are the sum of two squares. Explore the behaviour of (i.e. find good upper and lower bounds for) the consecutive differences $n_{k+1}-n_k$. | Erd\H{o}s \cite{Er51} proved that, for infinitely many $k$, $ n_{k+1}-n_k \gg \frac{\log n_k}{\sqrt{\log\log n_k}}. $ Richards \cite{Ri82} improved this to $ \limsup_{k\to \infty} \frac{n_{k+1}-n_k}{\log n_k} \geq 1/4. $ The constant $1/4$ here has been improved, most lately to $0.868\cdots$ by Dietmann, Elsholtz, Kal... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,999 | EP-233 | Erdős Problem #233 | Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Prove that $ \sum_{1\leq n\leq N}d_n^2 \ll N(\log N)^2. $ | Cramer \cite{Cr36} proved an upper bound of $O(N(\log N)^4)$ conditional on the Riemann hypothesis. Selberg \cite{Se43} improved this slightly (still assuming the Riemann hypothesis) to $ \sum_{1\leq n\leq N}\frac{d_n^2}{n}\ll (\log N)^4. $ The prime number theorem immediately implies a lower bound of $ \sum_{1\leq n\l... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
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"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,000 | EP-234 | Erdős Problem #234 | For every $c\geq 0$ the density $f(c)$ of integers for which $ \frac{p_{n+1}-p_n}{\log n}< c $ exists and is a continuous function of $c$.
",
"difficulty": "L1"
},{ | <!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** Existence and continuity of the limiting distribution of normalized consecutive-prime gaps remain open unconditionally. A sufficiently uniform Hardy-Littlewo... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,001 | EP-236 | Erdős Problem #236 | Let $f(n)$ count the number of solutions to $n=p+2^k$ for prime $p$ and $k\geq 0$. Is it true that $f(n)=o(\log n)$? | Erd\H{o}s \cite{Er50} proved that there are infinitely many $n$ such that $f(n)\gg \log\log n$.
Erd\H{o}s could not even prove that there do not exist infinitely many integers $n$ such that for all $1< 2^k<n$ the number $n-2^k$ is prime - he conjectured (see problem A19 of Guy's collection \cite{Gu04}) that $ 4,7,15,21... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,002 | EP-238 | Erdős Problem #238 | Let $c_1,c_2>0$. Is it true that, for any sufficiently large $x$, there exist more than $c_1\log x$ many consecutive primes $\leq x$ such that the difference between any two is $>c_2$? | Erd\H{o}s \cite{Er49c} proved this is true for any $c_2>0$ if $c_1>0$ is sufficiently small (depending on $c_1$).
References
[Er49c] Erd\H{o}s, P., On some applications of {B}run's method. Acta Univ. Szeged. Sect. Sci. Math. (1949), 57--63.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature ... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
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"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,003 | EP-241 | Erdős Problem #241 | Let $f(N)$ be the maximum size of $A\subseteq \{1,\ldots,N\}$ such that the sums $a+b+c$ with $a,b,c\in A$ are all distinct (aside from the trivial coincidences). Is it true that $ f(N)\sim N^{1/3}? $ | Originally asked to Erd\H{o}s by Bose. Bose and Chowla \cite{BoCh62} provided a construction proving one half of this, namely $ (1+o(1))N^{1/3}\leq f(N). $ The best upper bound known to date is due to Green \cite{Gr01}, $ f(N) \leq ((7/2)^{1/3}+o(1))N^{1/3} $ (note that $(7/2)^{1/3}\approx 1.519$).
More generally, Bose... | 1 | open | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,004 | EP-243 | Erdős Problem #243 | Let $1\leq a_1<a_2<\cdots$ be a sequence of integers such that $ \lim_{n\to \infty}\frac{a_n}{a_{n-1}^2}=1 $ and $\sum\frac{1}{a_n}\in \mathbb{Q}$. Then, for all sufficiently large $n\geq 1$, $ a_n = a_{n-1}^2-a_{n-1}+1. $ | Erd\H{o}s and Straus \cite{ErSt64} proved that if $\lim a_n/a_{n-1}^2=1$ and $\sum \frac{1}{a_n}$ is rational, and $a_n$ does not satisfy the recurrence, then $ \limsup_{n\to \infty} \frac{[a_1,\ldots,a_n]}{a_{n+1}}\left(\frac{a_n^2}{a_{n+1}}-1\right)>0. $ A sequence satisfying the reucrrence $a_n = a_{n-1}^2-a_{n-1}+1... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
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"slug": "number-theory",
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"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,005 | EP-244 | Erdős Problem #244 | Let $C>1$. Does the set of integers of the form $p+\lfloor C^k\rfloor$, for some prime $p$ and $k\geq 0$, have density $>0$? | Originally asked to Erd\H{o}s by Kalm\'{a}r. Erd\H{o}s believed the answer is yes. Romanoff \cite{Ro34} proved that the answer is yes if $C$ is an integer.
Ding \cite{Di25} has proved that this is true for almost all $C>1$.
References
[Di25] Y. Ding, On a Romanoff type problem of Erd\H{o}s and Kalm\'{a}r. arXiv:2503.... | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,006 | EP-247 | Erdős Problem #247 | Let $1\leq a_1<a_2<\cdots$ be a sequence of integers such that $ \limsup \frac{a_n}{n}=\infty. $ Is $ \sum_{n=1}^\infty \frac{1}{2^{a_n}} $ transcendental? | Erd\H{o}s \cite{Er75c} proved the answer is yes under the stronger condition that $\limsup n_k/k^t=\infty$ for all $t\geq 1$.
Erd\H{o}s \cite{Er88c} says 'many of these problems seem hopeless at present, but perhaps one can prove that if $a_n>cn^2$ then $\sum_{n=1}^\infty \frac{1}{2^{a_n}}$ is not the root of any quadr... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,007 | EP-249 | Erdős Problem #249 | Is $ \sum_n \frac{\phi(n)}{2^n} $ irrational? Here $\phi$ is the Euler totient function. | The decimal expansion of this sum is A256936 on the OEIS.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** Irrationality of sum phi(n)/2^n remains open. The current tracker rep... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,008 | EP-251 | Erdős Problem #251 | Is $ \sum \frac{p_n}{2^n} $ irrational? (Here $p_n$ is the $n$th prime.) | Erd\H{o}s \cite{Er58b} proved that $\sum \frac{p_n^k}{n!}$ is irrational for every $k\geq 1$.
In \cite{Er88c} he further conjectures that $\sum \frac{p_n^k}{2^n}$ is irrational for every $k$, and that if $g_n\geq 2$ and $g_n=o(p_n)$ then $ \sum_{n=1}^\infty \frac{p_n}{g_1\cdots g_n} $ is irrational. (The example $g_n=p... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,009 | EP-252 | Erdős Problem #252 | Let $k\geq 1$ and $\sigma_k(n)=\sum_{d\mid n}d^k$. Is $ \sum \frac{\sigma_k(n)}{n!} $ irrational? | This is known now for $1\leq k\leq 4$. The cases $k=1,2$ are reasonably straightforward, as observed by Erd\H{o}s \cite{Er52}. The case $k=3$ was proved independently by Schlage-Puchta \cite{ScPu06} and Friedlander, Luca, and Stoiciu \cite{FLC07}. The case $k=4$ was proved by Pratt \cite{Pr22}.
It is known that this su... | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,010 | EP-254 | Erdős Problem #254 | Let $A\subseteq \mathbb{N}$ be such that $ \lvert A\cap [1,2x]\rvert -\lvert A\cap [1,x]\rvert \to \infty\textrm{ as }x\to \infty $ and $ \sum_{n\in A} \{ \theta n\}=\infty $ for every $\theta\in (0,1)$, where $\{x\}$ is the distance of $x$ from the nearest integer. Then every sufficiently large integer is the sum of d... | Cassels \cite{Ca60} proved this under the alternative hypotheses $ \lim \frac{\lvert A\cap [1,2x]\rvert -\lvert A\cap [1,x]\rvert}{\log\log x}=\infty $ and $ \sum_{n\in A} \{ \theta n\}^2=\infty $ for every $\theta\in (0,1)$.
References
[Ca60] Cassels, J. W. S., On the representation of integers as the sums of distin... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,011 | EP-256 | Erdős Problem #256 | Let $n\geq 1$ and $f(n)$ be maximal such that for any integers $1\leq a_1\leq \cdots \leq a_n$ we have $ \max_{\lvert z\rvert=1}\left\lvert \prod_{i}(1-z^{a_i})\right\rvert\geq f(n). $ Estimate $f(n)$ - in particular, is it true that there exists some constant $c>0$ such that $ \log f(n) \gg n^c? $ | Erd\H{o}s and Szekeres \cite{ErSz59} proved that $\lim f(n)^{1/n}=1$ and $f(n)>\sqrt{2n}$. Erd\H{o}s proved an upper bound of $\log f(n) \ll n^{1-c}$ for some constant $c>0$ with probabilistic methods. Atkinson \cite{At61} showed that $\log f(n) \ll n^{1/2}\log n$.
This was improved to $ \log f(n) \ll n^{1/3}(\log n)^{... | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,012 | EP-257 | Erdős Problem #257 | Let $A\subseteq \mathbb{N}$ be an infinite set. Is $ \sum_{n\in A}\frac{1}{2^n-1} $ irrational? | If $A=\mathbb{N}$ then this series is $\sum_{n}\frac{d(n)}{2^n}$, where $d(n)$ is the number of divisors of $n$, which Erd\H{o}s \cite{Er48} proved is irrational. In general, if $f_A(n)$ counts the number of divisors of $n$ which are elements of $A$ then $ \sum_{n\in A}\frac{1}{2^n-1}=\sum_n \frac{f_A(n)}{2^n}. $ The c... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,013 | EP-258 | Erdős Problem #258 | Let $a_1,a_2,\ldots$ be a sequence of positive integers with $a_n\to \infty$. Is $ \sum_{n} \frac{\tau(n)}{a_1\cdots a_n} $ irrational, where $\tau(n)$ is the number of divisors of $n$? | Erd\H{o}s and Straus \cite{ErSt71} proved this is true if $a_n$ is monotone, i.e. $a_{n-1}\leq a_n$ for all $n$. Erd\H{o}s \cite{Er48} proved that $\sum_n \frac{d(n)}{t^n}$ is irrational for any integer $t\geq 2$.
Erd\H{o}s and Straus further conjectured that if $a_{n-1}\leq a_n$ for all $n$ then $ \sum_{n} \frac{\phi(... | 1 | solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | The maintained Erdős Problems source lists problem 258 as “proved (Lean)” on 2026-09-27. The source marks a Lean verification of the resolution.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://www.erdosproblems.com/258",
"label": "Erdős Problems #258: proved (Lean)"
},
{
"url": "https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml",
"label": "Erdős Problems machine-readable status list (snapshot checked 2026-09-27)"
},
{
"url": "https://huggin... | 2026-09-27T00:00:00 | Erdős Problems #258: proved (Lean): https://www.erdosproblems.com/258
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27T00:00:00",
"kind": "resolution",
"evidence": "maintained_source"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,014 | EP-260 | Erdős Problem #260 | Let $a_1<a_2<\cdots$ be an increasing sequence such that $a_n/n\to \infty$. Is the sum $ \sum_n \frac{a_n}{2^{a_n}} $ irrational? | Erd\H{o}s \cite{Er81l} proved this is true under either of the stronger assumptions that
{UL}
{LI} $a_{n+1}-a_n\to \infty$ or {/LI}
{LI} $a_n \gg n\sqrt{\log n\log\log n}$.{/LI}
{/UL}
Erd\H{o}s and Graham speculate that the condition $\limsup a_{n+1}-a_n=\infty$ is not sufficient, but know of no example.
References
[... | 1 | open | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,015 | EP-261 | Erdős Problem #261 | Are there infinitely many $n$ such that there exists some $t\geq 2$ and distinct integers $a_1,\ldots,a_t\geq 1$ such that $ \frac{n}{2^n}=\sum_{1\leq k\leq t}\frac{a_k}{2^{a_k}}? $ Is this true for all $n$? Is there a rational $x$ such that $ x = \sum_{k=1}^\infty \frac{a_k}{2^{a_k}} $ has at least $2^{\aleph_0}$ solu... | Related to [260].
In \cite{Er88c} Erd\H{o}s notes that Cusick had a simple proof that there do exist infinitely many such $n$. Erd\H{o}s does not record what this was, but a later paper by Borwein and Loring \cite{BoLo90} provides the following proof: for every positive integer $m$ and $n=2^{m+1}-m-2$ we have $ \frac{n... | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,016 | EP-263 | Erdős Problem #263 | Let $a_n$ be a sequence of positive integers such that for every sequence of positive integers $b_n$ with $b_n/a_n\to 1$ the sum $ \sum\frac{1}{b_n} $ is irrational. Is $a_n=2^{2^n}$ such a sequence? Must such a sequence satisfy $a_n^{1/n}\to \infty$? | One possible definition of an 'irrationality sequence' (see also [262] and [264]). A folklore result states that $\sum \frac{1}{a_n}$ is irrational whenever $\lim a_n^{1/2^n}=\infty$.
Kova\v{c} and Tao \cite{KoTa24} have proved that any strictly increasing sequence such that $\sum \frac{1}{a_n}$ converges and $\lim a_{... | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,017 | EP-264 | Erdős Problem #264 | Let $a_n$ be a sequence of positive integers such that for every bounded sequence of integers $b_n$ (with $a_n+b_n
eq 0$ and $b_n
eq 0$ for all $n$) the sum $ \sum \frac{1}{a_n+b_n} $ is irrational. Are $a_n=2^n$ or $a_n=n!$ examples of such a sequence? | A possible definition of an 'irrationality sequence' (see also [262] and [263]). One example is $a_n=2^{2^n}$. In \cite{ErGr80} they also ask whether such a sequence can have polynomial growth, but Erd\H{o}s later retracted this in \cite{Er88c}, claiming 'It is not hard to show that it cannot increase slower than expon... | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,018 | EP-265 | Erdős Problem #265 | Let $1\leq a_1<a_2<\cdots$ be an increasing sequence of integers. How fast can $a_n\to \infty$ grow if $ \sum\frac{1}{a_n}\quad\textrm{and}\quad\sum\frac{1}{a_n-1} $ are both rational? | Cantor observed that $a_n=\binom{n}{2}$ is such a sequence. If we replace $-1$ by a different constant then higher degree polynomials can be used - for example if we consider $\sum_{n\geq 2}\frac{1}{a_n}$ and $\sum_{n\geq 2}\frac{1}{a_n-12}$ then $a_n=n^3+6n^2+5n$ is an example of both series being rational.
Erd\H{o}s ... | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,019 | EP-267 | Erdős Problem #267 | Let $F_1=F_2=1$ and $F_{n+1}=F_n+F_{n-1}$ be the Fibonacci sequence. Let $n_1<n_2<\cdots $ be an infinite sequence with $n_{k+1}/n_k \geq c>1$. Must $ \sum_k\frac{1}{F_{n_k}} $ be irrational? | It may be sufficient to have $n_k/k\to \infty$. Good \cite{Go74} and Bicknell and Hoggatt \cite{BiHo76} have shown that $\sum \frac{1}{F_{2^n}}$ is irrational - in fact, $ \sum \frac{1}{F_{2^n}}=\frac{7-\sqrt{5}}{2}. $ Badea \cite{Ba87} proved that $\sum \frac{1}{F_{2^n+1}}$ is irrational.
The sum $\sum \frac{1}{F_n}$ ... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,020 | EP-269 | Erdős Problem #269 | Let $P$ be a finite set of primes with $\lvert P\rvert \geq 2$ and let $\{a_1<a_2<\cdots\}=\{ n\in \mathbb{N} : \textrm{if }p\mid n\textrm{ then }p\in P\}$. Is the sum $ \sum_{n=1}^\infty \frac{1}{[a_1,\ldots,a_n]}, $ where $[a_1,\ldots,a_n]$ is the lowest common multiple of $a_1,\ldots,a_n$, irrational? | If $P$ is infinite this sum is always irrational (in \cite{Er88c} Erd\H{o}s says this is a 'simple exercise').
This problem was asked by Erd\H{o}s in a letter to the editor written January 1st 1973 in issue 12 of the Fibonacci Quarterly, 1974, p. 335. In that letter he says that he can prove the sum is irrational if du... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,021 | EP-271 | Erdős Problem #271 | Let $A(n)=\{a_0<a_1<\cdots\}$ be the sequence defined by $a_0=0$ and $a_1=n$, and for $k\geq 1$ define $a_{k+1}$ as the least positive integer such that there is no three-term arithmetic progression in $\{a_0,\ldots,a_{k+1}\}$.
Can the $a_k$ be explicitly determined? How fast do they grow? | It is easy to see that $A(1)$ is the set of integers which have no 2 in their base 3 expansion. Odlyzko and Stanley \cite{OdSt78} have found similar characterisations are known for $A(3^k)$ and $A(2\cdot 3^k)$ for any $k\geq 0$ and conjectured in general that such a sequence always eventually either satisfies $ a_k\asy... | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,022 | EP-272 | Erdős Problem #272 | Let $N\geq 1$. What is the largest $t$ such that there are $A_1,\ldots,A_t\subseteq \{1,\ldots,N\}$ with $A_i\cap A_j$ a non-empty arithmetic progression for all $i
eq j$? | Simonovits and S\'{o}s \cite{SiSo81} have shown that $t\ll N^2$.
Erd\H{o}s and Graham asked whether the maximal $t$ is achieved when we take the $A_i$ to be all arithmetic progressions in $\{1,\ldots,N\}$ containing some fixed element, 'presumably the integer $\lfloor N/2\rfloor$'. This was disproved by Simonovits and ... | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,023 | EP-273 | Erdős Problem #273 | Is there a covering system all of whose moduli are of the form $p-1$ for some primes $p\geq 5$? | Selfridge has found an example using divisors of $360$ if $p=3$ is allowed.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** The existence of a covering system whose every modu... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,024 | EP-274 | Erdős Problem #274 | If $G$ is a group then can there exist an exact covering of $G$ by more than one cosets of different sizes? (i.e. each element is contained in exactly one of the cosets) | A question of Herzog and Sch"{onheim}, who conjectured more generally that if $G$ is any (not necessarily finite) group and $a_1G_1,\ldots,a_kG_k$ are finitely many cosets of subgroups of $G$ with distinct indices $[G:G_i]$ then the $a_iG_i$ cannot form a partition of $G$.
This conjecture was proved in the case when al... | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,025 | EP-276 | Erdős Problem #276 | Is there an infinite Lucas sequence $a_0,a_1,\ldots$ where $a_{n+2}=a_{n+1}+a_n$ for $n\geq 0$ such that all $a_k$ are composite, and yet no integer has a common factor with every term of the sequence? | Whether such a composite Lucas sequence even exists was open for a while, but using covering systems Graham \cite{Gr64} showed that $ a_0 = 1786772701928802632268715130455793 $ and $ a_1 = 1059683225053915111058165141686995 $ generate such a sequence. This problem asks whether one can have a composite Lucas sequence wi... | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,026 | EP-278 | Erdős Problem #278 | Let $A=\{n_1<\cdots<n_r\}$ be a finite set of positive integers. What is the maximum density of integers covered by a suitable choice of congruences $a_i\pmod{n_i}$?
Is the minimum density achieved when all the $a_i$ are equal? | Simpson \cite{Si86} has observed that the density of integers covered is at least $ \sum_i \frac{1}{n_i}-\sum_{i<j}\frac{1}{[n_i,n_j]}+\sum_{i<j<k}\frac{1}{[n_i,n_j,n_k]}-\cdot $ (where $[\cdots]$ denotes the least common multiple) which is achieved when all $a_i$ are equal, settling the second question.
References
[... | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | Maximum-density half: an exact characterisation and a fixed-parameter exact algorithm (2^{O(r^2)} poly(input)); the minimum half was settled by Simpson (1986). Whether an exact algorithm counts as an answer to 'what is the maximum density' is for the maintainers to judge. Alper Ferudun (2026), unrefereed preprint with ... | [
{
"url": "https://doi.org/10.5281/zenodo.22244392",
"label": "Alper Ferudun (2026), An Exact Fixed-Parameter Algorithm for Extremal Unions of Residue Classes — unrefereed preprint"
},
{
"url": "https://eulersolve.org/papers/ep-278/",
"label": "EulerSolve paper page and verification materials"
... | 2026-09-27T00:00:00 | null | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27T00:00:00",
"kind": "partial_progress",
"evidence": "unrefereed_preprint"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,027 | EP-279 | Erdős Problem #279 | Let $k\geq 3$. Is there a choice of congruence classes $a_p\pmod{p}$ for every prime $p$ such that all sufficiently large integers can be written as $a_p+tp$ for some prime $p$ and integer $t\geq k$? | Even the case $k=3$ seems difficult. This may be true with the primes replaced by any set $A\subseteq \mathbb{N}$ such that $ \lvert A\cap [1,N]\rvert \gg N/\log N $ and $ \sum_{\substack{n\in A\\ n\leq N}}\frac{1}{n} -\log\log N\to \infty $ as $N\to \infty$.
For $k=1$ or $k=2$ any set $A$ such that $\sum_{n\in A}\frac... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,028 | EP-281 | Erdős Problem #281 | Let $n_1<n_2<\cdots$ be an infinite sequence such that, for any choice of congruence classes $a_i\pmod{n_i}$, the set of integers not satisfying any of the congruences $a_i\pmod{n_i}$ has density $0$.
Is it true that for every $\epsilon>0$ there exists some $k$ such that, for every choice of congruence classes $a_i$, t... | The latter condition is clearly sufficient, the problem is if it's also necessary. The assumption implies $\sum \frac{1}{n_i}=\infty$. If the $n_i$ are pairwise relatively prime then it is sufficient that $\sum \frac{1}{n_i}=\infty$.
This is true - a proof is given in the comments by Somani (using ChatGPT).
An alternat... | 1 | solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | The maintained Erdős Problems source lists problem 281 as “proved (Lean)” on 2026-09-27. The source marks a Lean verification of the resolution.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://www.erdosproblems.com/281",
"label": "Erdős Problems #281: proved (Lean)"
},
{
"url": "https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml",
"label": "Erdős Problems machine-readable status list (snapshot checked 2026-09-27)"
},
{
"url": "https://huggin... | 2026-09-27T00:00:00 | Erdős Problems #281: proved (Lean): https://www.erdosproblems.com/281
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27T00:00:00",
"kind": "resolution",
"evidence": "maintained_source"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,029 | EP-282 | Erdős Problem #282 | Let $A\subseteq \mathbb{N}$ be an infinite set and consider the following greedy algorithm for a rational $x\in (0,1)$: choose the minimal $n\in A$ such that $n\geq 1/x$ and repeat with $x$ replaced by $x-\frac{1}{n}$. If this terminates after finitely many steps then this produces a representation of $x$ as the sum of... | In 1202 Fibonacci observed that this process terminates for any $x$ when $A=\mathbb{N}$. The problem when $A$ is the set of odd numbers is due to Stein.
Graham \cite{Gr64b} has shown that $\frac{m}{n}$ is the sum of distinct unit fractions with denominators $\equiv a\pmod{d}$ if and only if $ \left(\frac{n}{(n,(a,d))},... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,030 | EP-283 | Erdős Problem #283 | Let $p:\mathbb{Z}\to \mathbb{Z}$ be a polynomial whose leading coefficient is positive and such that there exists no $d\geq 2$ with $d\mid p(n)$ for all $n\geq 1$. Is it true that, for all sufficiently large $m$, there exist integers $1\leq n_1<\cdots <n_k$ such that $ 1=\frac{1}{n_1}+\cdots+\frac{1}{n_k} $ and $ m=p(n... | Graham \cite{Gr63} has proved this when $p(x)=x$. Graham also conjectures that this remains true with $1$ replaced by an arbitrary rational $\alpha>0$ (provided $m$ is taken sufficiently large depending on $\alpha$).
Cassels \cite{Ca60} has proved that these conditions on the polynomial imply every sufficiently large i... | 1 | solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | The maintained Erdős Problems source lists problem 283 as “proved (Lean)” on 2026-09-27. The source marks a Lean verification of the resolution.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://www.erdosproblems.com/283",
"label": "Erdős Problems #283: proved (Lean)"
},
{
"url": "https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml",
"label": "Erdős Problems machine-readable status list (snapshot checked 2026-09-27)"
},
{
"url": "https://huggin... | 2026-09-27T00:00:00 | Erdős Problems #283: proved (Lean): https://www.erdosproblems.com/283
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27T00:00:00",
"kind": "resolution",
"evidence": "maintained_source"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,031 | EP-288 | Erdős Problem #288 | Is it true that there are only finitely many pairs of intervals $I_1,I_2$ such that $ \sum_{n_1\in I_1}\frac{1}{n_1}+\sum_{n_2\in I_2}\frac{1}{n_2}\in \mathbb{N}? $ | For example, $ \frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{20}=1. $ This is still open even if $\lvert I_2\rvert=1$. It is perhaps true with two intervals replaced by any $k$ intervals.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** ope... | 1 | open | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,032 | EP-289 | Erdős Problem #289 | Is it true that, for all sufficiently large $k$, there exist finite intervals $I_1,\ldots,I_k\subset \mathbb{N}$, distinct, not overlapping or adjacent, with $\lvert I_i\rvert \geq 2$ for $1\leq i\leq k$ such that $ 1=\sum_{i=1}^k \sum_{n\in I_i}\frac{1}{n}? $ | Erd\H{o}s and Graham posed this in \cite{ErGr80} without the stipulation the intervals be distinct, non-overlapping, or adjacent, but Kovac in the comments has provided a simple argument showing that it is easily possible without this restriction, and likely \cite{ErGr80} just forgot to mention this natural restriction... | 1 | open | null | null | 3 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,033 | EP-291 | Erdős Problem #291 | Let $n\geq 1$ and define $L_n$ to be the least common multiple of $\{1,\ldots,n\}$ and $a_n$ by $ \sum_{1\leq k\leq n}\frac{1}{k}=\frac{a_n}{L_n}. $ Is it true that $(a_n,L_n)=1$ and $(a_n,L_n)>1$ both occur for infinitely many $n$? | Steinerberger has observed that the answer to the second question is trivially yes: for example, any $n$ which begins with a $2$ in base $3$ has $3\mid (a_n,L_n)$.
More generally, if the leading digit of $n$ in base $p$ is $p-1$ then $p\mid (a_n,L_n)$. There is in fact a necessary and sufficient condition: a prime $p\l... | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,034 | EP-293 | Erdős Problem #293 | Let $k\geq 1$ and let $v(k)$ be the minimal integer which does not appear as some $n_i$ in a solution to $ 1=\frac{1}{n_1}+\cdots+\frac{1}{n_k} $ with $1\leq n_1<\cdots <n_k$. Estimate the growth of $v(k)$. | Results of Bleicher and Erd\H{o}s \cite{BlEr75} imply $v(k) \gg k!$. It may be that $v(k)$ grows doubly exponentially in $\sqrt{k}$ or even $k$.
An elementary inductive argument shows that $n_k\leq ku_k$ where $u_1=1$ and $u_{i+1}=u_i(u_i+1)$, and hence $ v(k) \leq kc_0^{2^k}, $ where $ c_0=\lim_n u_n^{1/2^n}=1.26408\c... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,035 | EP-295 | Erdős Problem #295 | Let $N\geq 1$ and let $k(N)$ denote the smallest $k$ such that there exist $N\leq n_1<\cdots <n_k$ with $ 1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}. $ Is it true that $ \lim_{N\to \infty} k(N)-(e-1)N=\infty? $ | Erd\H{o}s and Straus \cite{ErSt71b} have proved the existence of some constant $c>0$ such that $ -c < k(N)-(e-1)N \ll \frac{N}{\log N}. $
References
[ErSt71b] Erd\H{o}s, P. and Straus, E. G., Solution to Problem. Amer. Math. Monthly (1971), 302-303.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## L... | 1 | open | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,036 | EP-301 | Erdős Problem #301 | Let $f(N)$ be the size of the largest $A\subseteq \{1,\ldots,N\}$ such that there are no solutions to $ \frac{1}{a}= \frac{1}{b_1}+\cdots+\frac{1}{b_k} $ with distinct $a,b_1,\ldots,b_k\in A$?
Estimate $f(N)$. In particular, is it true that $f(N)=(\tfrac{1}{2}+o(1))N$? | The example $A=(N/2,N]\cap \mathbb{N}$ shows that $f(N)\geq N/2$.
Wouter van Doorn has given an elementary argument that proves $ f(N)\leq (25/28+o(1))N. $ Indeed, consider the sets $S_a=\{2a,3a,4a,6a,12a\}\cap [1,N]$ as $a$ ranges over all integers of the form $8^b9^cd$ with $(d,6)=1$. All such $S_a$ are disjoint and,... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,037 | EP-302 | Erdős Problem #302 | Let $f(N)$ be the size of the largest $A\subseteq \{1,\ldots,N\}$ such that there are no solutions to $ \frac{1}{a}= \frac{1}{b}+\frac{1}{c} $ with distinct $a,b,c\in A$?
Estimate $f(N)$. In particular, is $f(N)=(\tfrac{1}{2}+o(1))N$? | The colouring version of this is [303], which was solved by Brown and R"{o}dl \cite{BrRo91}. One can take either $A$ to be all odd integers in $[1,N]$ or all integers in $[N/2,N]$ to show $f(N)\geq (1/2+o(1))N$.
Wouter van Doorn has proved (see this note) that $ f(N) \leq (9/10+o(1))N. $ Stijn Cambie has observed that ... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,038 | EP-304 | Erdős Problem #304 | For integers $1\leq a<b$ let $N(a,b)$ denote the minimal $k$ such that there exist integers $1<n_1<\cdots<n_k$ with $ \frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}. $ Estimate $N(b)=\max_{1\leq a<b}N(a,b)$. Is it true that $N(b) \ll \log\log b$? | Erd\H{o}s \cite{Er50c} proved that $ \log\log b \ll N(b) \ll \frac{\log b}{\log\log b}. $ The upper bound was improved by Vose \cite{Vo85} to $ N(b) \ll \sqrt{\log b}. $ One can also investigate the average of $N(a,b)$ for fixed $b$, and it is known that $ \frac{1}{b}\sum_{1\leq a<b}N(a,b) \gg \log\log b. $ Related to ... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,039 | EP-306 | Erdős Problem #306 | Let $a/b\in \mathbb{Q}_{>0}$ with $b$ squarefree. Are there integers $1<n_1<\cdots<n_k$, each the product of two distinct primes, such that $ \frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}? $ | For $n_i$ the product of three distinct primes, this is true when $b=1$, as proved by Butler, Erd\H{o}s and Graham \cite{BEG15} (this paper is perhaps Erd\H{o}s' last paper, appearing 19 years after his death).
References
[BEG15] Butler, Steve and Erd\H{o}s, Paul and Graham, Ron, Egyptian fractions with each denomina... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,040 | EP-311 | Erdős Problem #311 | Let $\delta(N)$ be the minimal non-zero value of $\lvert 1-\sum_{n\in A}\frac{1}{n}\rvert$ as $A$ ranges over all subsets of $\{1,\ldots,N\}$. Is it true that $ \delta(N)=e^{-(c+o(1))N} $ for some constant $c\in (0,1)$? | It is trivial that $ \delta(N)\geq \frac{1}{[1,\ldots,N]}=e^{-(1+o(1))N}, $ where $[1,\ldots,N]$ is the least common multiple of $\{1,\ldots,N\}$.
The formulation in \cite{ErGr80} has the additional condition that $A$ contain no $S$ such that $\sum_{n\in S}\frac{1}{n}=1$, but Kovac in the comments has shown that the si... | 1 | open | null | null | 3 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,041 | EP-312 | Erdős Problem #312 | Does there exist some $c>0$ such that, for any $K>1$, whenever $A$ is a sufficiently large finite multiset of positive integers with $\sum_{n\in A}\frac{1}{n}>K$ there exists some $S\subseteq A$ such that $ 1-e^{-cK} < \sum_{n\in S}\frac{1}{n}\leq 1? $ | Erd\H{o}s and Graham knew this with $e^{-cK}$ replaced by $c/K^2$.
",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** The exponential approximation-to-one conclusion remains ope... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,042 | EP-313 | Erdős Problem #313 | Are there infinitely many solutions to $ \frac{1}{p_1}+\cdots+\frac{1}{p_k}=1-\frac{1}{m}, $ where $m\geq 2$ is an integer and $p_1<\cdots<p_k$ are distinct primes? | For example, $ \frac{1}{2}+\frac{1}{3}=1-\frac{1}{6} $ and $ \frac{1}{2}+\frac{1}{3}+\frac{1}{7}=1-\frac{1}{42}. $ It is clear that we must have $m=p_1\cdots p_k$, and hence in particular there is at most one solution for each $m$. The integers $m$ for which there is such a solution are known as primary pseudoperfect n... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,043 | EP-317 | Erdős Problem #317 | Is there some constant $c>0$ such that for every $n\geq 1$ there exists some $\delta_k\in \{-1,0,1\}$ for $1\leq k\leq n$ with $ 0< \left\lvert \sum_{1\leq k\leq n}\frac{\delta_k}{k}\right\rvert < \frac{c}{2^n}? $ Is it true that for sufficiently large $n$, for any $\delta_k\in \{-1,0,1\}$, $ \left\lvert \sum_{1\leq k\... | Inequality is obvious for the second claim, the problem is strict inequality. This fails for small $n$, for example $ \frac{1}{2}-\frac{1}{3}-\frac{1}{4}=-\frac{1}{12}. $ Arguments of Kovac and van Doorn in the comment section prove a weak version of the first question, with an upper bound of $ 2^{-n\frac{(\log\log\log... | 1 | open | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,044 | EP-318 | Erdős Problem #318 | Let $A\subseteq \mathbb{N}$ be an infinite arithmetic progression and $f:A\to \{-1,1\}$ be a non-constant function. Must there exist a finite non-empty $S\subset A$ such that $ \sum_{n\in S}\frac{f(n)}{n}=0? $ What about if $A$ is an arbitrary set of positive density? What if $A$ is the set of squares excluding $1$? | Erd\H{o}s and Straus \cite{ErSt75} proved this when $A=\mathbb{N}$. Sattler \cite{Sa75} proved this when $A$ is the set of odd numbers. For the squares $1$ must be excluded or the result is trivially false, since $ \sum_{k\geq 2}\frac{1}{k^2}<1. $ This is false for some sets $A$ of positive density - indeed, it fails f... | 1 | solved | null | null | 3 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | The maintained Erdős Problems source lists problem 318 as “solved (Lean)” on 2026-09-27. The source marks a Lean verification of the resolution.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://www.erdosproblems.com/318",
"label": "Erdős Problems #318: solved (Lean)"
},
{
"url": "https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml",
"label": "Erdős Problems machine-readable status list (snapshot checked 2026-09-27)"
},
{
"url": "https://huggin... | 2026-09-27T00:00:00 | Erdős Problems #318: solved (Lean): https://www.erdosproblems.com/318
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27T00:00:00",
"kind": "resolution",
"evidence": "maintained_source"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,045 | EP-319 | Erdős Problem #319 | What is the size of the largest $A\subseteq \{1,\ldots,N\}$ such that there is a function $\delta:A\to \{-1,1\}$ such that $ \sum_{n\in A}\frac{\delta_n}{n}=0 $ and $ \sum_{n\in A'}\frac{\delta_n}{n}
eq 0 $ for all non-empty $A'\subsetneq A$? | Adenwalla has observed that a lower bound of $ \lvert A\rvert\geq (1-\tfrac{1}{e}+o(1))N $ follows from the main result of Croot \cite{Cr01}, which states that there exists some set of integers $B\subset [(\frac{1}{e}-o(1))N,N]$ such that $\sum_{b\in B}\frac{1}{b}=1$. Since the sum of $\frac{1}{m}$ for $m\in [c_1N,c_2N... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,046 | EP-320 | Erdős Problem #320 | Let $S(N)$ count the number of distinct sums of the form $\sum_{n\in A}\frac{1}{n}$ for $A\subseteq \{1,\ldots,N\}$. Estimate $S(N)$. | Bleicher and Erd\H{o}s \cite{BlEr75} proved the lower bound $ \log S(N)\geq \frac{N}{\log N}\left(\log 2\prod_{i=3}^k\log_iN\right), $ valid for $k\geq 4$ and $\log_kN\geq k$, and also \cite{BlEr76b} proved the upper bound $ \log S(N)\leq \frac{N}{\log N}\left(\log_r N \prod_{i=3}^r \log_iN\right), $ valid for $r\geq 1... | 1 | solved | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | The maintained Erdős Problems source lists problem 320 as “solved” on 2026-09-27. No Lean verification is inferred from the existence of a formalized statement.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://www.erdosproblems.com/320",
"label": "Erdős Problems #320: solved"
},
{
"url": "https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml",
"label": "Erdős Problems machine-readable status list (snapshot checked 2026-09-27)"
},
{
"url": "https://huggingface.c... | 2026-09-27T00:00:00 | Erdős Problems #320: solved: https://www.erdosproblems.com/320
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27T00:00:00",
"kind": "resolution",
"evidence": "maintained_source"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,047 | EP-321 | Erdős Problem #321 | What is the size of the largest $A\subseteq \{1,\ldots,N\}$ such that all sums $\sum_{n\in S}\frac{1}{n}$ are distinct for $S\subseteq A$? | Let $R(N)$ be the maximal such size. Results of Bleicher and Erd\H{o}s from \cite{BlEr75} and \cite{BlEr76b} imply that $ \frac{N}{\log N}\prod_{i=3}^k\log_iN\leq R(N)\leq \frac{1}{\log 2}\log_r N\left(\frac{N}{\log N} \prod_{i=3}^r \log_iN\right), $ valid for any $k\geq 4$ with $\log_kN\geq k$ and any $r\geq 1$ with $... | 1 | solved | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | The maintained Erdős Problems source lists problem 321 as “solved” on 2026-09-27. No Lean verification is inferred from the existence of a formalized statement.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://www.erdosproblems.com/321",
"label": "Erdős Problems #321: solved"
},
{
"url": "https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml",
"label": "Erdős Problems machine-readable status list (snapshot checked 2026-09-27)"
},
{
"url": "https://huggingface.c... | 2026-09-27T00:00:00 | Erdős Problems #321: solved: https://www.erdosproblems.com/321
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27T00:00:00",
"kind": "resolution",
"evidence": "maintained_source"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,048 | EP-322 | Erdős Problem #322 | Let $k\geq 3$ and $A\subset \mathbb{N}$ be the set of $k$th powers. What is the order of growth of $1_A^{(k)}(n)$, i.e. the number of representations of $n$ as the sum of $k$ many $k$th powers? Does there exist some $c>0$ and infinitely many $n$ such that $ 1_A^{(k)}(n) >n^c? $ | Connected to Waring's problem. The famous Hypothesis $K$ of Hardy and Littlewood was that $1_A^{(k)}(n)\leq n^{o(1)}$, but this was disproved by Mahler \cite{Ma36} for $k=3$, who constructed infinitely many $n$ such that $ 1_A^{(3)}(n)\gg n^{1/12} $ (where $A$ is the set of cubes). Erd\H{o}s believed Hypothesis $K$ fai... | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,049 | EP-323 | Erdős Problem #323 | Let $1\leq m\leq k$ and $f_{k,m}(x)$ denote the number of integers $\leq x$ which are the sum of $m$ many nonnegative $k$th powers. Is it true that $ f_{k,k}(x) \gg_\epsilon x^{1-\epsilon} $ for all $\epsilon>0$? Is it true that if $m<k$ then $ f_{k,m}(x) \gg x^{m/k} $ for sufficiently large $x$? | This would have significant applications to Waring's problem. Erd\H{o}s and Graham describe this as 'unattackable by the methods at our disposal'. The case $k=2$ was resolved by Landau, who showed $ f_{2,2}(x) \sim \frac{cx}{\sqrt{\log x}} $ for some constant $c>0$.
For $k>2$ it is not known if $f_{k,k}(x)=o(x)$.",
... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,050 | EP-324 | Erdős Problem #324 | Does there exist a polynomial $f(x)\in\mathbb{Z}[x]$ such that all the sums $f(a)+f(b)$ with $a<b$ nonnegative integers are distinct? | Erd\H{o}s and Graham describe this problem as 'very annoying'. Probably $f(x)=x^5$ should work. The Lander, Parkin, and Selfridge conjecture would imply that $f(x)=x^n$ has this property for all $n\geq 5$.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status... | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,051 | EP-325 | Erdős Problem #325 | Let $k\geq 3$ and $f_{k,3}(x)$ denote the number of integers $\leq x$ which are the sum of three nonnegative $k$th powers. Is it true that $ f_{k,3}(x) \gg x^{3/k} $ or even $\gg_\epsilon x^{3/k-\epsilon}$? | Mahler and Erd\H{o}s \cite{ErMa38} proved that $f_{k,2}(x) \gg x^{2/k}$. For $k=3$ the best known is due to Wooley \cite{Wo15}, $ f_{3,3}(x) \gg x^{0.917\cdots}. $ This problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.
References
[ErMa38] Erd\H{o}s, P\'{a}l and Mahler, Kur... | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,052 | EP-326 | Erdős Problem #326 | Let $A\subset \mathbb{N}$ be an additive basis of order $2$. Must there exist $B=\{b_1<b_2<\cdots\}\subseteq A$ which is also a basis such that $ \lim_{k\to \infty}\frac{b_k}{k^2} $ does not exist? | Erd\H{o}s originally asked whether this was true with $A=B$, but this was disproved by Cassels \cite{Ca57}.
This problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.
References
[Ca57] Cassels, J. W. S., "{U}ber Basen der nat"{u}rlichen Zahlenreihe. Abh. Math. Sem. Univ. Hambu... | 1 | partially_solved | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,053 | EP-327 | Erdős Problem #327 | Suppose $A\subseteq \{1,\ldots,N\}$ is such that if $a,b\in A$ and $a
eq b$ then $a+b
mid ab$. Can $A$ be 'substantially more' than the odd numbers?
What if $a,b\in A$ with $a
eq b$ implies $a+b
mid 2ab$? Must $\lvert A\rvert=o(N)$? | The connection to unit fractions comes from the observation that $\frac{1}{a}+\frac{1}{b}$ is a unit fraction if and only if $a+b\mid ab$.
Wouter van Doorn has given an elementary argument that proves that if $A\subseteq \{1,\ldots,N\}$ has $\lvert A\rvert \geq (25/28+o(1))N$ then $A$ must contain $a
eq b$ with $a+b\mi... | 1 | open | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,054 | EP-329 | Erdős Problem #329 | Suppose $A\subseteq \mathbb{N}$ is a Sidon set. How large can $ \limsup_{N\to \infty}\frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/2}} $ be? | Erd\H{o}s proved that $1/2$ is possible and Kr"{u}ckeberg \cite{Kr61} proved $1/\sqrt{2}$ is possible. Erd\H{o}s and Tur\'{a}n \cite{ErTu41} have proved this $\limsup$ is always $\leq 1$.
The fact that $1$ is possible would follow if any finite Sidon set is a subset of a perfect difference set (see [44] and [707]).
Thi... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,055 | EP-330 | Erdős Problem #330 | Does there exist a minimal basis with positive density, say $A\subset\mathbb{N}$, such that for any $n\in A$ the (upper) density of integers which cannot be represented without using $n$ is positive? | Asked by Erd\H{o}s and Nathanson.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-09-27)
**Status:** solved
**Classification:** SOLVED-IN-LITERATURE
**Current literature assessment.** The maintained Erdős Problems source lists problem 330 as “proved (Lean)” on 2026-09... | 1 | solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | The maintained Erdős Problems source lists problem 330 as “proved (Lean)” on 2026-09-27. The source marks a Lean verification of the resolution.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://www.erdosproblems.com/330",
"label": "Erdős Problems #330: proved (Lean)"
},
{
"url": "https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml",
"label": "Erdős Problems machine-readable status list (snapshot checked 2026-09-27)"
},
{
"url": "https://huggin... | 2026-09-27T00:00:00 | Erdős Problems #330: proved (Lean): https://www.erdosproblems.com/330
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27T00:00:00",
"kind": "resolution",
"evidence": "maintained_source"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,056 | EP-332 | Erdős Problem #332 | Let $A\subseteq \mathbb{N}$ and $D(A)$ be the set of those numbers which occur infinitely often as $a_1-a_2$ with $a_1,a_2\in A$. What conditions on $A$ are sufficient to ensure $D(A)$ has bounded gaps? | Prikry, Tijdeman, Stewart, and others (see the survey articles \cite{St78} and \cite{Ti79}) have shown that a sufficient condition is that $A$ has positive density.
One can also ask what conditions are sufficient for $D(A)$ to have positive density, or for $\sum_{d\in D(A)}\frac{1}{d}=\infty$, or even just $D(A)
eq\emp... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,057 | EP-334 | Erdős Problem #334 | Find the best function $f(n)$ such that every $n$ can be written as $n=a+b$ where both $a,b$ are $f(n)$-smooth (that is, are not divisible by any prime $p>f(n)$.) | Erd\H{o}s originally asked if even $f(n)\leq n^{1/3}$ is true. This is known, and the best bound is due to Balog \cite{Ba89} who proved that $ f(n) \ll_\epsilon n^{\frac{4}{9\sqrt{e}}+\epsilon} $ for all $\epsilon>0$. (Note $\frac{4}{9\sqrt{e}}=0.2695\ldots$.)
It is likely that $f(n)\leq n^{o(1)}$, or even $f(n)\leq e^... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,058 | EP-335 | Erdős Problem #335 | Let $d(A)$ denote the density of $A\subseteq \mathbb{N}$. Characterise those $A,B\subseteq \mathbb{N}$ with positive density such that $ d(A+B)=d(A)+d(B). $ | One way this can happen is if there exists $\theta>0$ such that $ A=\{ n>0 : \{ n\theta\} \in X_A\}\textrm{ and }B=\{ n>0 : \{n\theta\} \in X_B\} $ where $\{x\}$ denotes the fractional part of $x$ and $X_A,X_B\subseteq \mathbb{R}/\mathbb{Z}$ are such that $\mu(X_A+X_B)=\mu(X_A)+\mu(X_B)$. Are all possible $A$ and $B$ g... | 1 | partially_solved | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,059 | EP-336 | Erdős Problem #336 | For $r\geq 2$ let $h(r)$ be the maximal finite $k$ such that there exists a basis $A\subseteq \mathbb{N}$ of order $r$ (so every large integer is the sum of at most $r$ integers from $A$) and exact order $k$ (so every large integer is the sum of exactly $k$ integers from $A$).
Find the value of $ \lim_r \frac{h(r)}{r^2... | A simple example of the order of a basis differing from the exact order is given by $A=\cup_{k\geq 0}(2^{2k},2^{2k+1}]$, which has order $2$ but exact order $3$.
Erd\H{o}s and Graham \cite{ErGr80b} have shown that a basis $A$ has an exact order if and only if $a_2-a_1,a_3-a_2,a_4-a_3,\ldots$ are coprime. They also prov... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,060 | EP-338 | Erdős Problem #338 | The restricted order of a basis is the least integer $t$ (if it exists) such that every large integer is the sum of at most $t$ distinct summands from $A$. What are necessary and sufficient conditions that this exists? Can it be bounded (when it exists) in terms of the order of the basis? What are necessary and suffici... | Bateman has observed that for $h\geq 3$ there is a basis of order $h$ with no restricted order, taking $ A=\{1\}\cup \{x>0 : h\mid x\}. $ Kelly \cite{Ke57} has shown that any basis of order $2$ has restricted order at most $4$ and conjectured it always has restricted order at most $3$ (which he proved under the additio... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,061 | EP-340 | Erdős Problem #340 | Let $A=\{1,2,4,8,13,21,31,45,66,81,97,\ldots\}$ be the greedy Sidon sequence: we begin with $1$ and iteratively include the next smallest integer that preserves the Sidon property (i.e. there are no non-trivial solutions to $a+b=c+d$). What is the order of growth of $A$? Is it true that $ \lvert A\cap \{1,\ldots,N\}\rv... | This sequence is sometimes called the Mian-Chowla sequence. It is trivial that this sequence grows at least like $\gg N^{1/3}$.
Erd\H{o}s and Graham \cite{ErGr80} also asked about the difference set $A-A$, whether this has positive density, and whether this contains $22$. It does contain $22$, since $a_{15}-a_{14}=204-... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,062 | EP-341 | Erdős Problem #341 | Let $A=\{a_1<\cdots<a_k\}$ be a finite set of positive integers and extend it to an infinite sequence $\overline{A}=\{a_1<a_2<\cdots \}$ by defining $a_{n+1}$ for $n\geq k$ to be the least integer exceeding $a_n$ which is not of the form $a_i+a_j$ with $i,j\leq n$. Is it true that the sequence of differences $a_{m+1}-a... | An old problem of Dickson. Even a starting set as small as $\{1,4,9,16,25\}$ requires thousands of terms before periodicity occurs.
This problem is discussed under Problem 7 on Green's open problems list.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,063 | EP-342 | Erdős Problem #342 | With $a_1=1$ and $a_2=2$ let $a_{n+1}$ for $n\geq 2$ be the least integer $>a_n$ which can be expressed uniquely as $a_i+a_j$ for $i<j\leq n$.
What can be said about this sequence? Do infinitely many pairs $a,a+2$ occur? Does this sequence eventually have periodic differences? Is the density $0$? | A problem of Ulam. The sequence is $ 1,2,3,4,6,8,11,13,16,18,26,28,\ldots $ at OEIS A002858.
See also Problem 7 of Green's open problems list.
This is problem C4 in Guy's collection \cite{Gu04}.
References
[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.",
"difficulty": "L1"
},{
<!-... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
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