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|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2,064 | EP-345 | Erdős Problem #345 | Let $A\subseteq \mathbb{N}$ be a complete sequence, and define the threshold of completeness $T(A)$ to be the least integer $m$ such that all $n\geq m$ are in $ P(A) = \left\{\sum_{n\in B}n : B\subseteq A\textrm{ finite }\right\} $ (the existence of $T(A)$ is guaranteed by completeness).
Is it true that there are infin... | Erd\H{o}s and Graham \cite{ErGr80} remark that very little is known about $T(A)$ in general. It is known that $ T(n)=1, T(n^2)=128, T(n^3)=12758, $ $ T(n^4)=5134240,\textrm{ and }T(n^5)=67898771. $ Erd\H{o}s and Graham remark that a good candidate for the $n$ in the question are $k=2^t$ for large $t$, perhaps even $t=... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
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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 |
2,065 | EP-346 | Erdős Problem #346 | Let $A=\{1\leq a_1< a_2<\cdots\}$ be a set of integers such that
{UL}
{LI} $A\backslash B$ is complete for any finite subset $B$ and {/LI}
{LI} $A\backslash B$ is not complete for any infinite subset $B$.{/LI}
{/UL}
(Here 'complete' means all sufficiently large integers can be written as a sum of distinct members of th... | Graham \cite{Gr64d} has shown that the sequence $a_n=F_n-(-1)^{n}$, where $F_n$ is the $n$th Fibonacci number, has these properties. Erd\H{o}s and Graham \cite{ErGr80} remark that it is easy to see that if $a_{n+1}/a_n>\frac{1+\sqrt{5}}{2}$ then the second property is automatically satisfied, and that it is not hard to... | 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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"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
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"created_at": "2026-07-31T15:26:25.670Z"
} | {
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"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"
} | {
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"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.",
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"or... | The maintained Erdős Problems source lists problem 346 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/346",
"label": "Erdős Problems #346: solved"
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{
"url": "https://huggingface.c... | 2026-09-27T00:00:00 | Erdős Problems #346: solved: https://www.erdosproblems.com/346
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | {
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"checked_at": "2026-09-27T00:00:00",
"kind": "resolution",
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} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,066 | EP-348 | Erdős Problem #348 | For what values of $0\leq m<n$ is there a complete sequence $A=\{a_1\leq a_2\leq \cdots\}$ of integers such that
{UL}
{LI} $A$ remains complete after removing any $m$ elements, but {/LI}
{LI} $A$ is not complete after removing any $n$ elements? {/LI}
{/UL} | The Fibonacci sequence $1,1,2,3,5,\ldots$ shows that $m=1$ and $n=2$ is possible. The sequence of powers of $2$ shows that $m=0$ and $n=1$ is possible. The case $m=2$ and $n=3$ is not known.
van Doorn has shown that no such sequence exists for $2\leq m<n$ if we interpret complete in the strong sense that $ \left\{ \sum... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | 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... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,067 | EP-349 | Erdős Problem #349 | For what values of $t,\alpha \in (0,\infty)$ is the sequence $\lfloor t\alpha^n\rfloor$ complete (that is, all sufficiently large integers are the sum of distinct integers of the form $\lfloor t\alpha^n\rfloor$)? | Even in the range $t\in (0,1)$ and $\alpha\in (1,2)$ the behaviour is surprisingly complex. For example, Graham \cite{Gr64e} has shown that for any $k$ there exists some $t_k\in (0,1)$ such that the set of $\alpha$ such that the sequence is complete consists of at least $k$ disjoint line segments. It seems likely 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",
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"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 |
2,068 | EP-351 | Erdős Problem #351 | Let $p(x)\in \mathbb{Q}[x]$. Is it true that $ A=\{ p(n)+1/n : n\in \mathbb{N}\} $ is strongly complete, in the sense that, for any finite set $B$, $ \left\{\sum_{n\in X}n : X\subseteq A\backslash B\textrm{ finite }\right\} $ contains all sufficiently large integers? | Graham \cite{Gr63} proved this is true when $p(n)=n$. Erd\H{o}s and Graham also ask which rational functions $r(x)\in\mathbb{Z}(x)$ force $\{ r(n) : n\in\mathbb{N}\}$ to be complete?
Graham \cite{Gr64f} gave a complete characterisation of which polynomials $r\in \mathbb{R}[x]$ are such that $\{ r(n) : n\in \mathbb{N}\}... | 1 | solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 1,
"name": "number_theory",
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"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",
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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... | The maintained Erdős Problems source lists problem 351 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/351",
"label": "Erdős Problems #351: proved (Lean)"
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"url": "https://huggin... | 2026-09-27T00:00:00 | Erdős Problems #351: proved (Lean): https://www.erdosproblems.com/351
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | {
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"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 |
2,069 | EP-352 | Erdős Problem #352 | Is there some $c>0$ such that every measurable $A\subseteq \mathbb{R}^2$ of measure $\geq c$ contains the vertices of a triangle of area 1? | Erd\H{o}s (unpublished) proved that this is true if $A$ has infinite measure, or if $A$ is an unbounded set of positive measure (stating in \cite{Er78d} and \cite{Er83d} it 'follows easily from the Lebesgue density theorem').
In \cite{Er78d} and \cite{Er83d} he speculated that perhaps $C=4\pi/\sqrt{27}\approx 2.418$ wo... | 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 |
2,070 | EP-354 | Erdős Problem #354 | Let $\alpha,\beta\in \mathbb{R}_{>0}$ such that $\alpha/\beta$ is irrational. Is the multiset $ \{ \lfloor \alpha\rfloor,\lfloor 2\alpha\rfloor,\lfloor 4\alpha\rfloor,\ldots\}\cup \{ \lfloor \beta\rfloor,\lfloor 2\beta\rfloor,\lfloor 4\beta\rfloor,\ldots\} $ complete? That is, can all sufficiently large natural numbers... | This question was first mentioned by Graham \cite{Gr71}.
Hegyv\'{a}ri \cite{He89} proved that this holds if $\alpha=m/2^n$ is a dyadic rational and $\beta$ is not. He later \cite{He91} proved that, for any fixed $\alpha>0$, the set of $\beta$ for which this holds either has measure $0$ or infinite measure. In \cite{He9... | 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 |
2,071 | EP-357 | Erdős Problem #357 | Let $1\leq a_1<\cdots <a_k\leq n$ be integers such that all sums of the shape $\sum_{u\leq i\leq v}a_i$ are distinct. Let $f(n)$ be the maximal such $k$.
How does $f(n)$ grow? Is $f(n)=o(n)$? | Asked by Erd\H{o}s and Harzheim. In \cite{Er77c} Erd\H{o}s asks about an infinite such set of integers, and whether such a set must have density $0$. He notes that a simple averaging process implies $a_k \gg k\log k$ for infinitely many $k$, and so the lower density is $0$. He also asks whether $\sum\frac{1}{a_k}$ must... | 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 |
2,072 | EP-358 | Erdős Problem #358 | Let $A=\{a_1<\cdots\}$ be an infinite sequence of integers. Let $f(n)$ count the number of solutions to $ n=\sum_{u\leq i\leq v}a_i. $ Is there such an $A$ for which $f(n)\to \infty$ as $n\to \infty$? Or even where $f(n)\geq 2$ for all large $n$? | When $a_n=n$ the function $f(n)$ counts the number of odd divisors of $n$.
In modern language, this asks for the existence of a convex set $A$ such that $1_A\circ 1_A(n)\to \infty$ as $n\to \infty$.
Erd\H{o}s and Moser \cite{Mo63} considered the case when $A$ is the set of primes, and conjectured that the $\limsup$ of ... | 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",
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"slug": "erdos-problems",
"or... | The maintained Erdős Problems source lists problem 358 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/358",
"label": "Erdős Problems #358: proved (Lean)"
},
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{
"url": "https://huggin... | 2026-09-27T00:00:00 | Erdős Problems #358: proved (Lean): https://www.erdosproblems.com/358
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | {
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"checked_at": "2026-09-27T00:00:00",
"kind": "resolution",
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} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,073 | EP-359 | Erdős Problem #359 | Let $a_1<a_2<\cdots$ be an infinite sequence of integers such that $a_1=n$ and $a_{i+1}$ is the least integer which is not a sum of consecutive earlier $a_j$s. What can be said about the density of this sequence?
In particular, in the case $n=1$, can one prove that $a_k/k\to \infty$ and $a_k/k^{1+c}\to 0$ for any $c>0$... | A problem of MacMahon, studied by Andrews \cite{An75}. When $n=1$ this sequence begins $ 1,2,4,5,8,10,14,15,\ldots. $ This sequence is A002048 in the OEIS. Andrews conjectures $ a_k\sim \frac{k\log k}{\log\log k}. $ Porubsky \cite{Po77} proved that, for any $\epsilon>0$, there are infinitely many $k$ such that $ a_k < ... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
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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 |
2,074 | EP-361 | Erdős Problem #361 | Let $c>0$ and $n$ be some large integer. What is the size of the largest $A\subseteq \{1,\ldots,\lfloor cn\rfloor\}$ such that $n$ is not a sum of a subset of $A$? Does this depend on $n$ in an irregular way?
",
"difficulty": "L1"
},{ | <!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No verified general extremal theorem for the subset-sum-avoidance question was located.
**Verified partial progress.**
No distinct partial result was verif... | 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 |
2,075 | EP-365 | Erdős Problem #365 | Do all pairs of consecutive powerful numbers $n$ and $n+1$ come from solutions to Pell equations? In other words, must either $n$ or $n+1$ be a square?
Is the number of such $n\leq x$ bounded by $(\log x)^{O(1)}$? | Erd\H{o}s originally asked Mahler whether there are infinitely many pairs of consecutive powerful numbers, but Mahler immediately observed that the answer is yes from the infinitely many solutions to the Pell equation $x^2=2^3y^2+1$.
The list of $n$ such that $n$ and $n+1$ are both powerful is A060355 in the OEIS.
The ... | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
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"slug": "number-theory",
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"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"
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"id": 8,
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"slug": "erdos-problems",
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2,076 | EP-367 | Erdős Problem #367 | Let $B_2(n)$ be the 2-full part of $n$ (that is, $B_2(n)=n/n'$ where $n'$ is the product of all primes that divide $n$ exactly once). Is it true that, for every fixed $k\geq 1$, $ \prod_{n\leq m<n+k}B_2(m) \ll n^{2+o(1)}? $ Or perhaps even $\ll_k n^2$? | It would also be interesting to find upper and lower bounds for the analogous product with $B_r$ for $r\geq 3$, where $B_r(n)$ is the $r$-full part of $n$ (that is, the product of prime powers $p^a \mid n$ such that $p^{a+1}
mid n$ and $a\geq r$). Is it true that, for every fixed $r,k\geq 2$ and $\epsilon>0$, $ \limsup... | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
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"id": 1,
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"color_class": "text-green-600 bg-green-50 border-green-200"
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"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,077 | EP-368 | Erdős Problem #368 | How large is the largest prime factor of $n(n+1)$? | Let $F(n)$ be the prime in question. P\'{o}lya \cite{Po18} proved that $F(n)\to \infty$ as $n\to\infty$. Mahler \cite{Ma35} showed that $F(n)\gg \log\log n$. Schinzel \cite{Sc67b} observed that for infinitely many $n$ we have $F(n)\leq n^{O(1/\log\log\log n)}$.
The truth is probably $F(n)\gg (\log n)^2$ for all $n$. Er... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
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"slug": "erdos-problems",
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2,078 | EP-369 | Erdős Problem #369 | Let $\epsilon>0$ and $k\geq 2$. Is it true that, for all sufficiently large $n$, there is a sequence of $k$ consecutive integers in $\{1,\ldots,n\}$ all of which are $n^\epsilon$-smooth? | Erd\H{o}s and Graham state that this is open even for $k=2$ and 'the answer should be affirmative but the problem seems very hard'.
Unfortunately the problem is trivially true as written (simply taking $\{1,\ldots,k\}$ and $n>k^{1/\epsilon}$). There are (at least) two possible variants which are non-trivial, and it is ... | 1 | solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
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"or... | The maintained Erdős Problems source lists problem 369 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/369",
"label": "Erdős Problems #369: proved (Lean)"
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"url": "https://huggin... | 2026-09-27T00:00:00 | Erdős Problems #369: proved (Lean): https://www.erdosproblems.com/369
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | {
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"kind": "resolution",
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2,079 | EP-371 | Erdős Problem #371 | Let $P(n)$ denote the largest prime factor of $n$. Show that the set of $n$ with $P(n)<P(n+1)$ has density $1/2$. | Conjectured by Erd\H{o}s and Pomerance \cite{ErPo78}, who proved that this set and its complement both have positive upper density. The best unconditional lower bound available is due to L"{u} and Wang \cite{LuWa25}, who prove that $ \#\{ n<x :P(n)<P(n+1)\} > (0.2017-o(1))x, $ and the same lower bound for the complemen... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
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"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
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"id": 8,
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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 |
2,080 | EP-373 | Erdős Problem #373 | Show that the equation $ n! = a_1!a_2!\cdots a_k!, $ with $n-1>a_1\geq a_2\geq \cdots \geq a_k\geq 2$, has only finitely many solutions. | This would follow if $P(n(n+1))/\log n\to \infty$, where $P(m)$ denotes the largest prime factor of $m$ (see Problem [368]). Erd\H{o}s \cite{Er76d} proved that this problem would also follow from showing that $P(n(n-1))>4\log n$.
The condition $a_1<n-1$ is necessary to rule out the trivial solutions when $n=a_2!\cdots ... | 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,
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"id": 8,
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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 |
2,081 | EP-374 | Erdős Problem #374 | For any $m\in \mathbb{N}$, let $F(m)$ be the minimal $k\geq 2$ (if it exists) such that there are $a_1<\cdots <a_k=m$ with $a_1!\cdots a_k!$ a square. Let $D_k=\{ m : F(m)=k\}$. What is the order of growth of $\lvert D_k\cap\{1,\ldots,n\}\rvert$ for $3\leq k\leq 6$? For example, is it true that $\lvert D_6\cap \{1,\ldo... | Studied by Erd\H{o}s and Graham \cite{ErGr76} (see also \cite{LSS14}). It is known, for example, that:
{UL}
{LI}no $D_k$ contains a prime,{/LI}
{LI}$D_2=\{ n^2 : n>1\}$,{/LI}
{LI} $\lvert D_3\cap \{1,\ldots,n\}\rvert = o(\lvert D_4\cap \{1,\ldots,n\}\rvert)$,{/LI}
{LI} the least element of $D_6$ is $527$, and{/LI}
{LI}... | 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,
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"id": 8,
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"slug": "erdos-problems",
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2,082 | EP-376 | Erdős Problem #376 | Are there infinitely many $n$ such that $\binom{2n}{n}$ is coprime to $105$? | Erd\H{o}s, Graham, Ruzsa, and Straus \cite{EGRS75} have shown that, for any two odd primes $p$ and $q$, there are infinitely many $n$ such that $\binom{2n}{n}$ is coprime to $pq$.
This is equivalent (via Kummer's theorem) to whether there are infinitely many $n$ which have only digits $0,1$ in base $3$, digits $0,1,2$ ... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
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"id": 1,
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"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 |
2,083 | EP-377 | Erdős Problem #377 | Is there some absolute constant $C>0$ such that $ \sum_{p\leq n}1_{p
mid \binom{2n}{n}}\frac{1}{p}\leq C $ for all $n$ (where the summation is restricted to primes $p\leq n$)? | A question of Erd\H{o}s, Graham, Ruzsa, and Straus \cite{EGRS75}, who proved that if $f(n)$ is the sum in question then $ \lim_{x\to \infty}\frac{1}{x}\sum_{n\leq x}f(n) = \sum_{k=2}^\infty \frac{\log k}{2^k}=\gamma_0 $ and $ \lim_{x\to \infty}\frac{1}{x}\sum_{n\leq x}f(n)^2 = \gamma_0^2, $ so that for almost all integ... | 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,
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"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"
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"id": 8,
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"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,084 | EP-380 | Erdős Problem #380 | We call an interval $[u,v]$ 'bad' if the greatest prime factor of $\prod_{u\leq m\leq v}m$ occurs with an exponent greater than $1$. Let $B(x)$ count the number of $n\leq x$ which are contained in at least one bad interval. Is it true that $ B(x)\sim \#\{ n\leq x: P(n)^2\mid n\}, $ where $P(n)$ is the largest prime fac... | Erd\H{o}s and Graham only knew that $B(x) > x^{1-o(1)}$. Similarly, we call an interval $[u,v]$ 'very bad' if $\prod_{u\leq m\leq v}m$ is powerful. The number of integers $n\leq x$ contained in at least one very bad interval should be $\ll x^{1/2}$. In fact, it should be asymptotic to the number of powerful numbers $\l... | 1 | solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
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"slug": "erdos-problems",
"or... | The maintained Erdős Problems source lists problem 380 as “proved” 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/380",
"label": "Erdős Problems #380: proved"
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"url": "https://huggingface.c... | 2026-09-27T00:00:00 | Erdős Problems #380: proved: https://www.erdosproblems.com/380
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | {
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2,085 | EP-382 | Erdős Problem #382 | Let $u\leq v$ be such that the largest prime dividing $\prod_{u\leq m\leq v}m$ appears with exponent at least $2$. Is it true that $v-u=v^{o(1)}$? Can $v-u$ be arbitrarily large? | Erd\H{o}s and Graham report it follows from results of Ramachandra that $v-u\leq v^{1/2+o(1)}$.
Cambie has observed that the first question boils down to some old conjectures on prime gaps.
By Cram\'{er's conjecture}, for every $\epsilon>0,$ for every $u$ sufficiently large there is a prime between $u$ and $u+u^\epsilo... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
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"slug": "erdos-problems",
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2,086 | EP-383 | Erdős Problem #383 | Is it true that for every $k$ there are infinitely many primes $p$ such that the largest prime divisor of $ \prod_{0\leq i\leq k}(p^2+i) $ is $p$? | A positive answer to this would give an answer to the second part of [382]. Heuristically, the 'probability' that $n$ has no prime divisors $\geq n^{1/2}$ is $1-\log 2>0$, so standard heuristics predict the answer to this is yes.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (check... | 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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"slug": "number-theory",
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"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 |
2,087 | EP-385 | Erdős Problem #385 | Let $ F(n) = \max_{\substack{m<n\\ m\textrm{ composite}}} m+p(m), $ where $p(m)$ is the least prime divisor of $m$. Is it true that $F(n)>n$ for all sufficiently large $n$? Does $F(n)-n\to \infty$ as $n\to\infty$? | A question of Erd\H{o}s, Eggleton, and Selfridge, who write that 'plausible conjectures on primes' imply that $F(n)\leq n$ for only finitely many $n$, and in fact it is possible that this quantity is always at least $n+(1-o(1))\sqrt{n}$ (note that it is trivially $\leq n+\sqrt{n}$).
Tao has discussed this problem in 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",
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"id": 1,
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"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"
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"id": 8,
"name": "erdos_problems",
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"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,088 | EP-386 | Erdős Problem #386 | Let $2\leq k\leq n-2$. Can $\binom{n}{k}$ be the product of consecutive primes infinitely often? For example $ \binom{21}{2}=2\cdot 3\cdot 5\cdot 7. $ | Erd\H{o}s and Graham write that 'a proof that this cannot happen infinitely often for $\binom{n}{2}$ seems hopeless; probably this can never happen for $\binom{n}{k}$ if $3\leq k\leq n-3$.'
Weisenberg has provided four easy examples that show Erd\H{o}s and Graham were too optimistic here: $ \binom{7}{3}=5\cdot 7, $ $ ... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
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"id": 1,
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"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"
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"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 |
2,089 | EP-387 | Erdős Problem #387 | Is there an absolute constant $c>0$ such that, for all $1\leq k< n$, the binomial coefficient $\binom{n}{k}$ has a divisor in $(cn,n]$? | Erd\H{o}s once conjectured that $\binom{n}{k}$ must always have a divisor in $(n-k,n]$, but this was disproved by Schinzel and Erd\H{o}s \cite{Sc58}. A counterexample is given by $n=99215$ and $k=15$. Schinzel conjectured (see problem B34 of \cite{Gu04}) that, for all sufficiently large $k$ which are not prime powers, ... | 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 387 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/387",
"label": "Erdős Problems #387: 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 #387: solved: https://www.erdosproblems.com/387
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 |
2,090 | EP-388 | Erdős Problem #388 | Can one classify all solutions of $ \prod_{1\leq i\leq k_1}(m_1+i)=\prod_{1\leq j\leq k_2}(m_2+j) $ where $k_1,k_2>3$ and $m_1+k_1\leq m_2$? Are there only finitely many solutions? | More generally, if $k_1>2$ then for fixed $a$ and $b$ $ a\prod_{1\leq i\leq k_1}(m_1+i)=b\prod_{1\leq j\leq k_2}(m_2+j) $ should have only a finite number of solutions.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-T... | 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 |
2,091 | EP-389 | Erdős Problem #389 | Is it true that for every $n\geq 1$ there is a $k$ such that $ n(n+1)\cdots(n+k-1)\mid (n+k)\cdots (n+2k-1)? $ | Asked by Erd\H{o}s and Straus.
For example when $n=2$ we have $k=5$: $ 2\times 3 \times 4 \times 5\times 6 \mid 7 \times 8 \times 9\times 10\times 11. $ and when $n=3$ we have $k=4$: $ 3\times 4\times 5\times 6 \mid 7\times 8\times 9\times 10. $ Bhavik Mehta has computed the minimal such $k$ for $1\leq n\leq 18$ (now a... | 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 |
2,092 | EP-390 | Erdős Problem #390 | Let $f(n)$ be the minimal $m$ such that $ n! = a_1\cdots a_k $ with $n< a_1<\cdots <a_k=m$. Is there (and what is it) a constant $c$ such that $ f(n)-2n \sim c\frac{n}{\log n}? $ | Erd\H{o}s, Guy, and Selfridge \cite{EGS82} have shown that $ f(n)-2n \asymp \frac{n}{\log n}. $
References
[EGS82] Erd\H{o}s, P. and Guy, R. K. and Selfridge, J. L., Another property of {$239$} and some related questions. Congr. Numer. (1982), 243-257.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
#... | 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 |
2,093 | EP-393 | Erdős Problem #393 | Let $f(n)$ denote the minimal $m\geq 1$ such that $ n! = a_1\cdots a_t $ with $a_1<\cdots <a_t=a_1+m$. What is the behaviour of $f(n)$? | Erd\H{o}s and Graham write that they do not even know whether $f(n)=1$ infinitely often (i.e. whether a factorial is the product of two consecutive integers infinitely often).
Let $F_m(N)$ count the number of $n\leq N$ such that $f(n)=m$. Berend and Osgood \cite{BeOs92} proved that, for each fixed $m$, $F_m(N)=o(N)$. B... | 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 |
2,094 | EP-394 | Erdős Problem #394 | Let $t_k(n)$ denote the least $m$ such that $ n\mid m(m+1)(m+2)\cdots (m+k-1). $ Is it true that $ \sum_{n\leq x}t_2(n)\ll \frac{x^2}{(\log x)^c} $ for some $c>0$?
Is it true that, for $k\geq 2$, $ \sum_{n\leq x}t_{k+1}(n) =o\left(\sum_{n\leq x}t_k(n)\right)? $ | In \cite{ErGr80} they mention a conjecture of Erd\H{o}s that the sum is $o(x^2)$. This was proved by Erd\H{o}s and Hall \cite{ErHa78}, who proved that in fact $ \sum_{n\leq x}t_2(n)\ll \frac{\log\log\log x}{\log\log x}x^2. $ Erd\H{o}s and Hall conjecture that the sum is $o(x^2/(\log x)^c)$ for any $c<\log 2$.
Since $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 |
2,095 | EP-396 | Erdős Problem #396 | Is it true that for every $k$ there exists $n$ such that $ \prod_{0\leq i\leq k}(n-i) \mid \binom{2n}{n}? $ | Erd\H{o}s and Graham write that $n+1$ always divides $\binom{2n}{n}$ (indeed $\frac{1}{n+1}\binom{2n}{n}$ is the $n$th Catalan number), but it is quite rare that $n$ divides $\binom{2n}{n}$.
Pomerance \cite{Po14} has shown that for any $k\geq 0$ there are infinitely many $n$ such that $n-k\mid\binom{2n}{n}$, although 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 |
2,096 | EP-400 | Erdős Problem #400 | For any $k\geq 2$ let $g_k(n)$ denote the maximum value of $ (a_1+\cdots+a_k)-n $ where $a_1,\ldots,a_k$ are integers such that $a_1!\cdots a_k! \mid n!$. Can one show that $ \sum_{n\leq x}g_k(n) \sim c_k x\log x $ for some constant $c_k$? Is it true that there is a constant $c_k$ such that for almost all $n<x$ we have... | Erd\H{o}s and Graham write that it is easy to show that $g_k(n) \ll_k \log n$ always, but the best possible constant is unknown.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.*... | 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 |
2,097 | EP-404 | Erdős Problem #404 | For which integers $a\geq 1$ and primes $p$ is there a finite upper bound on those $k$ such that there are $a=a_1<\cdots<a_n$ with $ p^k \mid (a_1!+\cdots+a_n!)? $ If $f(a,p)$ is the greatest such $k$, how does this function behave?
Is there a prime $p$ and an infinite sequence $a_1<a_2<\cdots$ such that if $p^{m_k}$ i... | See also [403]. Lin \cite{Li76} has shown that $f(2,2) \leq 254$.
References
[Li76] Lin, S., On two problems of Erd\H{o}s concerning sums of distinct factorials. Bell Laboratories internal memorandum (1960).",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**St... | 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 |
2,098 | EP-406 | Erdős Problem #406 | Is it true that there are only finitely many powers of $2$ which have only the digits $0$ and $1$ when written in base $3$? | The only examples seem to be $1$, $4=1+3$, and $256=1+3+3^2+3^5$. If we only allow the digits $1$ and $2$ then $2^{15}$ seems to be the largest such power of $2$.
This would imply via Kummer's theorem that $ 3\mid \binom{2^{k+1}}{2^k} $ for all large $k$.
Saye \cite{Sa22} has computed that $2^n$ contains every possible... | 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 |
2,099 | EP-408 | Erdős Problem #408 | Let $\phi(n)$ be the Euler totient function and $\phi_k(n)$ be the iterated $\phi$ function, so that $\phi_1(n)=\phi(n)$ and $\phi_k(n)=\phi(\phi_{k-1}(n))$. Let $ f(n) = \min \{ k : \phi_k(n)=1\}. $ Does $f(n)/\log n$ have a distribution function? Is $f(n)/\log n$ almost always constant? What can be said about the lar... | Pillai \cite{Pi29} was the first to investigate this function, and proved $ \log_3 n < f(n) < \log_2 n $ for all large $n$. Shapiro \cite{Sh50} proved that $f(n)$ is essentially multiplicative.
Erd\H{o}s, Granville, Pomerance, and Spiro \cite{EGPS90} have proved that the answer to the first two questions is yes, condit... | 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 |
2,100 | EP-409 | Erdős Problem #409 | How many iterations of $n\mapsto \phi(n)+1$ are needed before a prime is reached? Can infinitely many $n$ reach the same prime? What is the density of $n$ which reach any fixed prime? | A problem of Finucane. One can also ask similar questions about $n\mapsto \sigma(n)-1$: do iterates of this always reach a prime? If so, how soon? (It is easily seen that iterates of this cannot reach the same prime infinitely often, since they are non-decreasing.)
This problem is somewhat ambiguously phrased. Let $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 |
2,101 | EP-410 | Erdős Problem #410 | Let $\sigma_1(n)=\sigma(n)$, the sum of divisors function, and $\sigma_k(n)=\sigma(\sigma_{k-1}(n))$. Is it true that for all $n\geq 2$ $ \lim_{k\to \infty} \sigma_k(n)^{1/k}=\infty? $ | This is discussed in problem B9 of Guy's collection \cite{Gu04}.
References
[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Cur... | 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 |
2,102 | EP-411 | Erdős Problem #411 | Let $g_1=g(n)=n+\phi(n)$ and $g_k(n)=g(g_{k-1}(n))$. For which $n$ and $r$ is it true that $g_{k+r}(n)=2g_k(n)$ for all large $k$? | The known solutions to $g_{k+2}(n)=2g_k(n)$ are $n=10$ and $n=94$. Selfridge and Weintraub found solutions to $g_{k+9}(n)=9g_k(n)$ and Weintraub found $ g_{k+25}(3114)=729g_k(3114) $ for all $k\geq 6$.
Steinerberger \cite{St25} has observed that, for $r=2$, this problem is equivalent to asking for solutions to $ \phi(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 |
2,103 | EP-412 | Erdős Problem #412 | Let $\sigma_1(n)=\sigma(n)$, the sum of divisors function, and $\sigma_k(n)=\sigma(\sigma_{k-1}(n))$.
Is it true that, for every $m,n\geq 2$, there exist some $i,j$ such that $\sigma_i(m)=\sigma_j(n)$? | In \cite{Er79d} Erd\H{o}s attributes this conjecture to van Wijngaarden, who told it to Erd\H{o}s in the 1950s.
That is, there is (eventually) only one possible sequence that the iterated sum of divisors function can settle on. Selfridge reports numerical evidence which suggests the answer is no, but Erd\H{o}s and Grah... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
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2,104 | EP-413 | Erdős Problem #413 | Let $\omega(n)$ count the number of distinct primes dividing $n$. Are there infinitely many $n$ such that, for all $m<n$, we have $m+\omega(m) \leq n$?
Can one show that there exists an $\epsilon>0$ such that there are infinitely many $n$ where $m+\epsilon \omega(m)\leq n$ for all $m<n$? | In \cite{Er79} Erd\H{o}s calls such an $n$ a 'barrier' for $\omega$. Some other natural number theoretic functions (such as $\phi$ and $\sigma$) have no barriers because they increase too rapidly. Erd\H{o}s believed that $\omega$ should have infinitely many barriers. In \cite{Er79d} he proves that $F(n)=\prod k_i$, whe... | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
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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.",
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"id": 8,
"name": "erdos_problems",
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"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,105 | EP-414 | Erdős Problem #414 | Let $h_1(n)=h(n)=n+\tau(n)$ (where $\tau(n)$ counts the number of divisors of $n$) and $h_k(n)=h(h_{k-1}(n))$. Is it true, for any $m,n$, there exist $i$ and $j$ such that $h_i(m)=h_j(n)$? | Asked by Spiro. That is, there is (eventually) only one possible sequence that the iterations of $n\mapsto h(n)$ can settle on. Erd\H{o}s and Graham believed the answer is yes. Similar questions can be asked by the iterates of many other functions. See also [412] and [413].",
"difficulty": "L1"
},{
<!-- LITERATURE... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
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"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,106 | EP-415 | Erdős Problem #415 | For any $n$ let $F(n)$ be the largest $k$ such that any of the $k!$ possible ordering patterns appears in some sequence of $\phi(m+1),\ldots,\phi(m+k)$ with $m+k\leq n$. Is it true that $ F(n)=(c+o(1))\log\log\log n $ for some constant $c$? Is the first pattern which fails to appear always $ \phi(m+1)>\phi(m+2)>\cdots ... | Erd\H{o}s \cite{Er36b} proved that $ F(n)\asymp \log\log\log n, $ and similarly if we replace $\phi$ with $\sigma$ or $\tau$ or $
u$ or any 'decent' additive or multiplicative function.
Weisenberg has observed that the same questions could be asked for ordering patterns which allow equality (indeed, the final problem o... | 1 | partially_solved | 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.",
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"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,107 | EP-416 | Erdős Problem #416 | Let $V(x)$ count the number of $n\leq x$ such that $\phi(m)=n$ is solvable. Does $V(2x)/V(x)\to 2$? Is there an asymptotic formula for $V(x)$? | Pillai \cite{Pi29} proved $V(x)=o(x)$. Erd\H{o}s \cite{Er35b} proved $V(x)=x(\log x)^{-1+o(1)}$.
The behaviour of $V(x)$ is now almost completely understood. Maier and Pomerance \cite{MaPo88} proved $ V(x)=\frac{x}{\log x}e^{(C+o(1))(\log\log\log x)^2}, $ for some explicit constant $C>0$. Ford \cite{Fo98} improved this... | 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"
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"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 |
2,108 | EP-417 | Erdős Problem #417 | Let $ V'(x)=\#\{\phi(m) : 1\leq m\leq x\} $ and $ V(x)=\#\{\phi(m) \leq x : 1\leq m\}. $ Does $\lim V(x)/V'(x)$ exist? Is it $>1$? | It is trivial that $V'(x) \leq V(x)$. In \cite{Er98} Erd\H{o}s suggests the limit may be infinite. See also [416].
References
[Er98] Erd\H{o}s, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180.",
"difficulty": "L1"
},{
<!-- LITERA... | 1 | open | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
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"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,109 | EP-420 | Erdős Problem #420 | If $\tau(n)$ counts the number of divisors of $n$ then let $ F(f,n)=\frac{\tau((n+\lfloor f(n)\rfloor)!)}{\tau(n!)}. $ Is it true that $ \lim_{n\to \infty}F((\log n)^C,n)=\infty $ for large $C$?
Is it true that $F(\log n,n)$ is everywhere dense in $(1,\infty)$?
More generally, if $f(n)\leq \log n$ is a monotonic functi... | Erd\H{o}s and Graham write that it is easy to show that $\lim F(n^{1/2},n)=\infty$, and in fact the $n^{1/2}$ can be replaced by $n^{1/2-c}$ for some small constant $c>0$.
Erd\H{o}s, Graham, Ivi\'{c}, and Pomerance \cite{EGIP96} have proved that $ \liminf F(c\log n, n) = 1 $ for any $c>0$, and $ \lim F(n^{4/9},n)=\inft... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
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"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,110 | EP-421 | Erdős Problem #421 | Is there a sequence $1\leq d_1<d_2<\cdots$ with density $1$ such that all products $\prod_{u\leq i\leq v}d_i$ are distinct? | A construction of Selfridge (see [786]) shows that there exists such a sequence of density $>1/e-\epsilon$ for any $\epsilon>0$.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-09-27)
**Status:** solved
**Classification:** SOLVED-IN-LITERATURE
**Current literature ass... | 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"
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"id": 1,
"level": 1,
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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... | The maintained Erdős Problems source lists problem 421 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/421",
"label": "Erdős Problems #421: 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 #421: solved: https://www.erdosproblems.com/421
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 |
2,111 | EP-422 | Erdős Problem #422 | Let $f(1)=f(2)=1$ and for $n>2$ $ f(n) = f(n-f(n-1))+f(n-f(n-2)). $ Does $f(n)$ miss infinitely many integers? What is its behaviour? | Asked by Hofstadter. The sequence begins $1,1,2,3,3,4,\ldots$ and is A005185 in the OEIS. It is not even known whether $f(n)$ is well-defined for all $n$.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Curre... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
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"name": "number_theory",
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"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
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"id": 8,
"name": "erdos_problems",
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"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,112 | EP-423 | Erdős Problem #423 | Let $a_1=1$ and $a_2=2$ and for $k\geq 3$ choose $a_k$ to be the least integer $>a_{k-1}$ which is the sum of at least two consecutive terms of the sequence. What is the asymptotic behaviour of this sequence? | Asked by Hofstadter (in \cite{Er77c} Erd\H{o}s says Hofstadter was inspired by a similar question of Ulam). The sequence begins $ 1,2,3,5,6,8,10,11,\ldots $ and is A005243 in the OEIS.
Bolan and Tang have independently proved that there are infinitely many integers which do not appear in this sequence. In fact, the seq... | 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",
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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",
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"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,113 | EP-424 | Erdős Problem #424 | Let $a_1=2$ and $a_2=3$ and continue the sequence by appending to $a_1,\ldots,a_n$ all possible values of $a_ia_j-1$ with $i
eq j$. Is it true that the set of integers which eventually appear has positive density? | Asked by Hofstadter. The sequence begins $2,3,5,9,14,17,26,\ldots$ and is A005244 in the OEIS. This problem is also discussed in section E31 of Guy's book Unsolved Problems in Number Theory.
In \cite{ErGr80} (and in Guy's book) this problem as written is asking for whether almost all integers appear in this sequence, b... | 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,
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"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,
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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 |
2,114 | EP-425 | Erdős Problem #425 | Let $F(n)$ be the maximum possible size of a subset $A\subseteq\{1,\ldots,N\}$ such that the products $ab$ are distinct for all $a<b$. Is there a constant $c$ such that $ F(n)=\pi(n)+(c+o(1))n^{3/4}(\log n)^{-3/2}? $ If $A\subseteq \{1,\ldots,n\}$ is such that all products $a_1\cdots a_r$ are distinct for $a_1<\cdots <... | Erd\H{o}s \cite{Er68} proved that there exist some constants $0<c_1\leq c_2$ such that $ \pi(n)+c_1 n^{3/4}(\log n)^{-3/2}\leq F(n)\leq \pi(n)+c_2 n^{3/4}(\log n)^{-3/2}. $ This problem can also be considered in the real numbers: that is, what is the size of the the largest $A\subset [1,x]$ such that for any distinct $... | 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,
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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 |
2,115 | EP-428 | Erdős Problem #428 | Is there a set $A\subseteq \mathbb{N}$ such that, for infinitely many $n$, all of $n-a$ are prime for all $a\in A$ with $0<a<n$ and $ \liminf\frac{\lvert A\cap [1,x]\rvert}{\pi(x)}>0? $ | Erd\H{o}s and Graham could show this is true (assuming the prime $k$-tuple conjecture) if we replace $\liminf$ by $\limsup$.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** Th... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
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"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"
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"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,116 | EP-430 | Erdős Problem #430 | Fix some integer $n$ and define a decreasing sequence in $[1,n)$ by $a_1=n-1$ and, for $k\geq 2$, letting $a_k$ be the greatest integer in $[1,a_{k-1})$ such that all of the prime factors of $a_k$ are $>n-a_k$.
Is it true that, for sufficiently large $n$, not all of this sequence can be prime? | Erd\H{o}s and Graham write 'preliminary calculations made by Selfridge indicate that this is the case but no proof is in sight'. For example if $n=8$ we have $a_1=7$ and $a_2=5$ and then must stop.
Sarosh Adenwalla has observed that this problem is equivalent to (the first part of) [385]. Indeed, assuming a positive an... | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
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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 |
2,117 | EP-431 | Erdős Problem #431 | Are there two infinite sets $A$ and $B$ such that $A+B$ agrees with the set of prime numbers up to finitely many exceptions? | A problem of Ostmann, sometimes known as the 'inverse Goldbach problem'. The answer is surely no. The best result in this direction is due to Elsholtz and Harper \cite{ElHa15}, who showed that if $A,B$ are such sets then for all large $x$ we must have $ \frac{x^{1/2}}{\log x\log\log x} \ll \lvert A \cap [1,x]\rvert \ll... | 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 |
2,118 | EP-432 | Erdős Problem #432 | Let $A,B\subseteq \mathbb{N}$ be two infinite sets. How dense can $A+B$ be if all elements of $A+B$ are pairwise relatively prime? | Asked by Straus, inspired by a problem of Ostmann (see [431]).",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PARTIAL-PROGRESS
**Current literature assessment.** A universal prime-counting upper bound is eleme... | 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 |
2,119 | EP-436 | Erdős Problem #436 | If $p$ is a prime and $k,m\geq 2$ then let $r(k,m,p)$ be the minimal $r$ such that $r,r+1,\ldots,r+m-1$ are all $k$th power residues modulo $p$. Let $ \Lambda(k,m)=\limsup_{p\to \infty} r(k,m,p). $ Is it true that $\Lambda(k,2)$ is finite for all $k$? Is $\Lambda(k,3)$ finite for all odd $k$? How large are they? | Asked by Lehmer and Lehmer \cite{LeLe62}, who note that for example $\Lambda(2,2)=9$ - indeed, $9$ is always a quadratic residue, and if $10$ isn't then either $2$ or $5$ is, and hence at least one of $1,2$ or $4,5$ or $9,10$ is a consecutive pair of quadratic residues (and similarly there are infinitely many $p$ for w... | 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 |
2,120 | EP-445 | Erdős Problem #445 | Is it true that, for any $c>1/2$, if $p$ is a sufficiently large prime then, for any $n\geq 0$, there exist $a,b\in(n,n+p^c)$ such that $ab\equiv 1\pmod{p}$? | Heilbronn (unpublished) proved this for $c$ sufficiently close to $1$. Heath-Brown \cite{He00} used Kloosterman sums to prove this for all $c>3/4$.
This is discussed in this MathOverflow question.
References
[He00] Heath-Brown, D. R., Arithmetic applications of {K}loosterman sums. Nieuw Arch. Wiskd. (5) (2000), 380--... | 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 |
2,121 | EP-450 | Erdős Problem #450 | How large must $y=y(\epsilon,n)$ be such that the number of integers in $(x,x+y)$ with a divisor in $(n,2n)$ is at most $\epsilon y$? | It is not clear what the intended quantifier on $x$ is. Cambie has observed that if this is intended to hold for all $x$ then, provided $ \epsilon(\log n)^\delta (\log\log n)^{3/2}\to \infty $ as $n\to \infty$, where $\delta=0.086\cdots$, there is no such $y$, which follows from an averaging argument and the work of Fo... | 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 |
2,122 | EP-451 | Erdős Problem #451 | Estimate $n_k$, the smallest integer $>2k$ such that $\prod_{1\leq i\leq k}(n_k-i)$ has no prime factor in $(k,2k)$. | Erd\H{o}s and Graham write 'we can prove $n_k>k^{1+c}$ but no doubt much more is true'.
In \cite{Er79d} Erd\H{o}s writes that probably $n_k<e^{o(k)}$ but $n_k>k^d$ for all constant $d$.
Adenwalla observes that an easy upper bound is $n_k\leq \prod_{k<p<2k}p=e^{O(k)}$.
References
[Er79d] Erd\H{o}s, P., Some unconventi... | 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 |
2,123 | EP-452 | Erdős Problem #452 | Let $\omega(n)$ count the number of distinct prime factors of $n$. What is the size of the largest interval $I\subseteq [x,2x]$ such that $\omega(n)>\log\log n$ for all $n\in I$? | Erd\H{o}s \cite{Er37} proved that the density of integers $n$ with $\omega(n)>\log\log n$ is $1/2$. The Chinese remainder theorem implies that there is such an interval with $ \lvert I\rvert \geq (1+o(1))\frac{\log x}{(\log\log x)^2}. $ It could be true that there is such an interval of length $(\log x)^{k}$ for arbitr... | 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 |
2,124 | EP-454 | Erdős Problem #454 | Let $ f(n) = \min_{i<n} (p_{n+i}+p_{n-i}), $ where $p_k$ is the $k$th prime. Is it true that $ \limsup_n (f(n)-2p_n)=\infty? $ | Pomerance \cite{Po79} has proved the $\limsup$ is at least $2$.
References
[Po79] Pomerance, Carl, The prime number graph. Math. Comp. (1979), 399-408.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Curren... | 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 |
2,125 | EP-455 | Erdős Problem #455 | Let $q_1<q_2<\cdots$ be a sequence of primes such that $ q_{n+1}-q_n\geq q_n-q_{n-1}. $ Must $ \lim_n \frac{q_n}{n^2}=\infty? $ | Richter \cite{Ri76} proved that $ \liminf_n \frac{q_n}{n^2}>0.352\cdots. $
References
[Ri76] Richter, Bernd, "{U}ber die Monotonie von Differenzenfolgen. Acta Arith. (1976), 225-227.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classif... | 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 |
2,126 | EP-456 | Erdős Problem #456 | Let $p_n$ be the smallest prime $\equiv 1\pmod{n}$ and let $m_n$ be the smallest integer such that $n\mid \phi(m_n)$.
Is it true that $m_n<p_n$ for almost all $n$? Does $p_n/m_n\to \infty$ for almost all $n$? Are there infinitely many primes $p$ such that $p-1$ is the only $n$ for which $m_n=p$? | Linnik's theorem implies that $p_n\leq n^{O(1)}$. It is trivial that $m_n\leq p_n$ always.
If $n=q-1$ for some prime $q$ then $m_n=p_n$. Erd\H{o}s \cite{Er79e} writes it is 'easy to show' that for infinitely many $n$ we have $m_n <p_n$, and that $m_n/n\to \infty$ for almost all $n$.
van Doorn in the comments has noted ... | 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 |
2,127 | EP-457 | Erdős Problem #457 | Is there some $\epsilon>0$ such that there are infinitely many $n$ where all primes $p\leq (2+\epsilon)\log n$ divide $ \prod_{1\leq i\leq \log n}(n+i)? $ | A problem of Erd\H{o}s and Pomerance.
More generally, let $q(n,k)$ denote the least prime which does not divide $\prod_{1\leq i\leq k}(n+i)$. This problem asks whether $q(n,\log n)\geq (2+\epsilon)\log n$ infinitely often. Taking $n$ to be the product of primes between $\log n$ and $(2+o(1))\log n$ gives an example whe... | 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 457 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/457",
"label": "Erdős Problems #457: 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 #457: proved (Lean): https://www.erdosproblems.com/457
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 |
2,128 | EP-460 | Erdős Problem #460 | Let $a_0=0$ and $a_1=1$, and in general define $a_k$ to be the least integer $>a_{k-1}$ for which $(n-a_k,n-a_i)=1$ for all $0\leq i<k$. Does $ \sum_{0<a_i< n}\frac{1}{a_i}\to \infty $ as $n\to \infty$? What about if we restrict the sum to those $i$ such that $n-a_j$ is divisible by some prime $\leq a_j$, or the comple... | This question arose in work of Eggleton, Erd\H{o}s, and Selfridge, who could prove that $a_k <k^{2+o(1)}$ for $k$ large enough depending on $n$, but conjectured that in fact $a_k\ll k\log k$ is true.
The problem above is from \cite{Er77c}. This question is stated slightly differently in \cite{ErGr80}, which has $a_0=n$... | 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 |
2,129 | EP-461 | Erdős Problem #461 | Let $s_t(n)$ be the $t$-smooth component of $n$ - that is, the product of all primes $p$ (with multiplicity) dividing $n$ such that $p<t$. Let $f(n,t)$ count the number of distinct possible values for $s_t(m)$ for $m\in [n+1,n+t]$. Is it true that $ f(n,t)\gg t $ (uniformly, for all $t$ and $n$)? | Erd\H{o}s and Graham report they can show $ f(n,t) \gg \frac{t}{\log t}. $ ",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** The uniform linear lower bound for the number of di... | 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 |
2,130 | EP-462 | Erdős Problem #462 | Let $p(n)$ denote the least prime factor of $n$. There is a constant $c>0$ such that $ \sum_{\substack{n<x\\ n\textrm{ not prime}}}\frac{p(n)}{n}\sim c\frac{x^{1/2}}{(\log x)^2}. $ Is it true that there exists a constant $C>0$ such that $ \sum_{x\leq n\leq x+Cx^{1/2}(\log x)^2}\frac{p(n)}{n} \gg 1 $ for all large $x$?
... | <!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** The stated least-prime-factor lower bound in every interval of length C sqrt(x)(log x)^2 remains open.
**Verified partial progress.**
- A global composite-... | 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 |
2,131 | EP-463 | Erdős Problem #463 | Is there a function $f$ with $f(n)\to \infty$ as $n\to \infty$ such that, for all large $n$, there is a composite number $m$ such that $ n+f(n)<m<n+p(m)? $ (Here $p(m)$ is the least prime factor of $m$.) | In \cite{Er92e} Erd\H{o}s asks about $ F(n)=\min_{m>n}(m-p(m)), $ and whether $n-F(n)\sim cn^{1/2}$ for some $c>0$.
See also [385].
References
[Er92e] Erd\H{o}s, P\'{a}l, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEG... | 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 |
2,132 | EP-467 | Erdős Problem #467 | Prove the following for all large $x$: there is a choice of congruence classes $a_p$ for all primes $p\leq x$ and a decomposition $\{p\leq x\}=A\sqcup B$ into two non-empty sets such that, for all $n<x$, there exist some $p\in A$ and $q\in B$ such that $n\equiv a_p\pmod{p}$ and $n\equiv a_q\pmod{q}$. | This is what I assume the intended problem is, although the presentation in \cite{ErGr80} is missing some crucial quantifiers, so I may have misinterpreted it.
References
[ErGr80] Erd\H{o}s, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique... | 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 |
2,133 | EP-468 | Erdős Problem #468 | For any $n$ let $D_n$ be the set of sums of the shape $d_1,d_1+d_2,d_1+d_2+d_3,\ldots$ where $1<d_1<d_2<\cdots$ are the divisors of $n$.
What is the size of $D_n\backslash \cup_{m<n}D_m$?
If $f(N)$ is the minimal $n$ such that $N\in D_n$ then is it true that $f(N)=o(N)$? Perhaps just for almost all $N$?",
"difficul... | <!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** The divisor-prefix-sum novelty and minimal-index questions remain open.
**Verified partial progress.**
No distinct partial result was verified beyond the s... | 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 |
2,134 | EP-469 | Erdős Problem #469 | Let $A$ be the set of all $n$ such that $n=d_1+\cdots+d_k$ with $d_i$ distinct proper divisors of $n$, but this is not true for any $m\mid n$ with $m<n$. Does $ \sum_{n\in A}\frac{1}{n} $ converge? | The integers in $A$ are also known as primitive pseudoperfect numbers and are listed as A006036 in the OEIS.
The same question can be asked for those $n$ which do not have distinct sums of sets of divisors, but any proper divisor of $n$ does (which are listed as A119425 in the OEIS).
Benkoski and Erd\H{o}s \cite{BeEr74... | 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 469 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/469",
"label": "Erdős Problems #469: 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 #469: proved (Lean): https://www.erdosproblems.com/469
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 |
2,135 | EP-470 | Erdős Problem #470 | Call $n$ weird if $\sigma(n)\geq 2n$ and $n$ is not pseudoperfect, that is, it is not the sum of any set of its divisors.
Are there any odd weird numbers? Are there infinitely many primitive weird numbers, i.e. those such that no proper divisor of $n$ is weird? | Weird numbers were investigated by Benkoski and Erd\H{o}s \cite{BeEr74}, who proved that the set of weird numbers has positive density. The smallest weird number is $70$.
Melfi \cite{Me15} has proved that there are infinitely many primitive weird numbers, conditional on the fact that $p_{n+1}-p_n<\frac{1}{10}p_n^{1/2}$... | 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 |
2,136 | EP-472 | Erdős Problem #472 | Given some initial finite sequence of primes $q_1<\cdots<q_m$ extend it so that $q_{n+1}$ is the smallest prime of the form $q_n+q_i-1$ for $n\geq m$. Is there an initial starting sequence so that the resulting sequence is infinite? | A problem due to Ulam. For example if we begin with $3,5$ then the sequence continues $3,5,7,11,13,17,\ldots$. It is possible that this sequence is infinite.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Cu... | 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 |
2,137 | EP-477 | Erdős Problem #477 | Is there a polynomial $f:\mathbb{Z}\to \mathbb{Z}$ of degree at least $2$ and a set $A\subset \mathbb{Z}$ such that for any $n\in \mathbb{Z}$ there is exactly one $a\in A$ and $b\in \{ f(n) : n\in\mathbb{Z}\}$ such that $n=a+b$? | A question of Erd\H{o}s and Graham, who thought the answer was negative.",
"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 pr... | 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 477 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/477",
"label": "Erdős Problems #477: 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 #477: solved: https://www.erdosproblems.com/477
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 |
2,138 | EP-478 | Erdős Problem #478 | Let $p$ be a prime and $ A_p = \{ k! \pmod{p} : 1\leq k<p\}. $ Is it true that $ \lvert A_p\rvert \sim (1-\tfrac{1}{e})p? $ | Since $A_p/A_p=\{1,\ldots,p-1\}$ it follows that $\lvert A_p\rvert \gg p^{1/2}$. The best known lower bound is due to Grebennikov, Sagdeev, Semchankau, and Vasilevskii \cite{GSSV24}, $ \lvert A_p\rvert \geq (\sqrt{2}-o(1))p^{1/2}, $ which follows from proving that $\lvert A_pA_p\rvert=(1+o(1))p$.
Wilson's theorem impli... | 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 |
2,139 | EP-479 | Erdős Problem #479 | Is it true that, for all $k
eq 1$, there are infinitely many $n$ such that $2^n\equiv k\pmod{n}$? | A conjecture of Graham. It is easy to see that $2^n
ot\equiv 1\mod{n}$ for all $n>1$, so the restriction $k
eq 1$ is necessary. Erd\H{o}s and Graham report that Graham, Lehmer, and Lehmer have proved this for $k=2^i$ for $i\geq 1$, or if $k=-1$, but I cannot find such a paper. Tang has written a short note giving a pro... | 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 |
2,140 | EP-483 | Erdős Problem #483 | Let $f(k)$ be the minimal $N$ such that if $\{1,\ldots,N\}$ is $k$-coloured then there is a monochromatic solution to $a+b=c$. Estimate $f(k)$. In particular, is it true that $f(k) < c^k$ for some constant $c>0$? | The values of $f(k)$ are known as Schur numbers. The best-known bounds for large $k$ are $ (380)^{k/5}-O(1)\leq f(k) \leq \lfloor(e-\tfrac{1}{24}) k!\rfloor-1. $ The lower bound is due to Ageron, Casteras, Pellerin, Portella, Rimmel, and Tomasik \cite{ACPPRT21} (improving previous bounds of Exoo \cite{Ex94} and Fredric... | 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 |
2,141 | EP-486 | Erdős Problem #486 | Let $A\subseteq \mathbb{N}$, and for each $n\in A$ choose some $X_n\subseteq \mathbb{Z}/n\mathbb{Z}$. Let $ B = \{ m\in \mathbb{N} : m
ot\in X_n\pmod{n}\textrm{ for all }n\in A\textrm{ with }m>n\}. $ Must $B$ have a logarithmic density, i.e. is it true that $ \lim_{x\to \infty} \frac{1}{\log x}\sum_{\substack{m\in B\\ ... | Davenport and Erd\H{o}s \cite{DaEr36} proved that the answer is yes when $X_n=\{0\}$ for all $n\in A$. An alternative elementary proof was later given by Davenport and Erd\H{o}s in \cite{DaEr51}.
The problem considers logarithmic density since Besicovitch \cite{Be34} showed examples exist without a natural density, eve... | 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 |
2,142 | EP-488 | Erdős Problem #488 | Let $A$ be a finite set and $ B=\{ n \geq 1 : a\mid n\textrm{ for some }a\in A\}. $ Is it true that, for every $m>n\geq \max(A)$, $ \frac{\lvert B\cap [1,m]\rvert }{m}< 2\frac{\lvert B\cap [1,n]\rvert}{n}? $ | The constant $2$ would be the best possible here, as witnessed by taking $A=\{a\}$, $n=2a-1$, and $m=2a$.
This problem is also discussed in problem E5 of Guy's collection \cite{Gu04}.
In \cite{Er61} this problem is as stated above, but with $a\mid n$ in the definition of $B$ replaced by $a
mid n$. This is most likely 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 |
2,143 | EP-489 | Erdős Problem #489 | Let $A\subseteq \mathbb{N}$ be a set such that $\lvert A\cap [1,x]\rvert=o(x^{1/2})$. Let $ B=\{ n\geq 1 : a
mid n\textrm{ for all }a\in A\}. $ If $B=\{b_1<b_2<\cdots\}$ then is it true that $ \lim \frac{1}{x}\sum_{b_i<x}(b_{i+1}-b_i)^2 $ exists (and is finite)? | For example, when $A=\{p^2: p\textrm{ prime}\}$ then $B$ is the set of squarefree numbers, and the existence of this limit was proved by Erd\H{o}s.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current lite... | 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 |
2,144 | EP-495 | Erdős Problem #495 | Let $\alpha,\beta \in \mathbb{R}$. Is it true that $ \liminf_{n\to \infty} n \| n\alpha \| \| n\beta\| =0 $ where $\|x\|$ is the distance from $x$ to the nearest integer? | The infamous Littlewood conjecture.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** Littlewood's conjecture remains open.
**Verified partial progress.**
- The conjecture is ... | 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 |
2,145 | EP-500 | Erdős Problem #500 | What is $\mathrm{ex}_3(n,K_4^3)$? That is, the largest number of $3$-edges which can placed on $n$ vertices so that there exists no $K_4^3$, a set of 4 vertices which is covered by all 4 possible $3$-edges. | A problem of Tur\'{a}n. Tur\'{a}n observed that dividing the vertices into three equal parts $X_1,X_2,X_3$, and taking the edges to be those triples that either have exactly one vertex in each part or two vertices in $X_i$ and one vertex in $X_{i+1}$ (where $X_4=X_1$) shows that $ \mathrm{ex}_3(n,K_4^3)\geq\left(\frac{... | 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 |
2,146 | EP-501 | Erdős Problem #501 | For every $x\in\mathbb{R}$ let $A_x\subset \mathbb{R}$ be a bounded set with outer measure $<1$. Must there exist an infinite independent set, that is, some infinite $X\subseteq \mathbb{R}$ such that $x
ot\in A_y$ for all $x
eq y\in X$?
If the sets $A_x$ are closed and have measure $<1$, then must there exist an indepe... | Erd\H{o}s and Hajnal \cite{ErHa60} proved the existence of arbitrarily large finite independent sets (under the assumptions in the first problem).
Gladysz \cite{Gl62} proved the existence of an independent set of size $2$ under the assumptions of the the second question.
Hechler \cite{He72} has shown the answer to the ... | 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 |
2,147 | EP-503 | Erdős Problem #503 | What is the size of the largest $A\subseteq \mathbb{R}^d$ such that every three points from $A$ determine an isosceles triangle? That is, for any three points $x,y,z$ from $A$, at least two of the distances $\lvert x-y\rvert,\lvert y-z\rvert,\lvert x-z\rvert$ are equal. | When $d=2$ the answer is $6$ (due to Kelly \cite{ErKe47} - an alternative proof is given by Kov\'{a}cs \cite{Ko24c}). When $d=3$ the answer is $8$ (due to Croft \cite{Cr62}). The best upper bound known in general is due to Blokhuis \cite{Bl84} who showed that $ \lvert A\rvert \leq \binom{d+2}{2}. $ Alweiss has observed... | 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 |
2,148 | EP-507 | Erdős Problem #507 | Let $\alpha(n)$ be such that every set of $n$ points in the unit disk contains three points which determine a triangle of area at most $\alpha(n)$. Estimate $\alpha(n)$. | Heilbronn's triangle problem. It is trivial that $\alpha(n) \ll 1/n$. Erd\H{o}s observed that $\alpha(n)\gg 1/n^2$. The current best bounds are $ \frac{\log n}{n^2}\ll \alpha(n) \ll \frac{1}{n^{7/6+o(1)}}. $ The lower bound is due to Koml\'{o}s, Pintz, and Szemer\'{e}di \cite{KPS82}. The upper bound is due to Cohen, Po... | 1 | partially_solved | null | null | 6 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"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 |
2,149 | EP-508 | Erdős Problem #508 | What is the chromatic number of the plane? That is, what is the smallest number of colours required to colour $\mathbb{R}^2$ such that no two points of the same colour are distance $1$ apart? | The Hadwiger-Nelson problem. Let $\chi$ be the chromatic number of the plane. An equilateral triangle trivially shows that $\chi\geq 3$. There are several small graphs that show $\chi\geq 4$ (in particular the Moser spindle and Golomb graph). The best bounds currently known are $ 5 \leq \chi \leq 7. $ The lower bound i... | 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 |
2,150 | EP-509 | Erdős Problem #509 | Let $f(z)\in\mathbb{C}[z]$ be a monic non-constant polynomial. Can the set $ \{ z\in \mathbb{C} : \lvert f(z)\rvert \leq 1\} $ be covered by a set of circles the sum of whose radii is $\leq 2$? | Cartan proved this is true with $2$ replaced by $2e$, which was improved to $2.59$ by Pommerenke \cite{Po61}. Pommerenke \cite{Po59} proved that $2$ is achievable if the set is connected (see [1046]).
The generalisation of this to higher dimensions was asked by Erd\H{o}s as Problem 4.23 in \cite{Ha74}.
References
[Ha... | 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 |
2,151 | EP-510 | Erdős Problem #510 | If $A\subset \mathbb{Z}$ is a finite set of size $N$ then is there some absolute constant $c>0$ and $\theta$ such that $ \sum_{n\in A}\cos(n\theta) < -cN^{1/2}? $ | Chowla's cosine problem. Ruzsa \cite{Ru04} (improving on an earlier result of Bourgain \cite{Bo86}), proved an upper bound of $ -\exp(O(\sqrt{\log N})). $ Polynomial bounds were proved independently by Bedert \cite{Be25c} and Jin, Milojevi\'{c}, Tomon, and Zhang \cite{JMTZ25}. The best bound follows from the method of ... | 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 |
2,152 | EP-513 | Erdős Problem #513 | Let $f=\sum_{n=0}^\infty a_nz^n$ be a transcendental entire function. What is the greatest possible value of $ \liminf_{r\to \infty} \frac{\max_n\lvert a_nr^n\rvert}{\max_{\lvert z\rvert=r}\lvert f(z)\rvert}? $ | It is trivial that this value is in $[1/2,1)$. K"{o}v\'{a}ri (unpublished) observed that it must be $>1/2$. Clunie and Hayman \cite{ClHa64} showed that it is $\leq 2/\pi-c$ for some absolute constant $c>0$. Some other results on this quantity were established by Gray and Shah \cite{GrSh63}.
See also [227].
References
... | 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 |
2,153 | EP-514 | Erdős Problem #514 | Let $f(z)$ be an entire transcendental function. Does there exist a path $L$ so that, for every $n$, $ \lvert f(z)/z^n\rvert \to \infty $ as $z\to \infty$ along $L$?
Can the length of this path be estimated in terms of $M(r)=\max_{\lvert z\rvert=r}\lvert f(z)\rvert$? Does there exist a path along which $\lvert f(z)\rve... | Boas (unpublished) has proved the first part, that such a path must exist.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PARTIAL-PROGRESS
**Current literature assessment.** The first path-existence component ... | 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 |
2,154 | EP-517 | Erdős Problem #517 | Let $f(z)=\sum_{k=1}^\infty a_kz^{n_k}$ be an entire function (with $a_k
eq 0$ for all $k\geq 1$). Is it true that if $n_k/k\to \infty$ then $f(z)$ assumes every value infinitely often? | A conjecture of Fej\'{e}r and P\'{o}lya.
Fej\'{e}r \cite{Fe08} proved that if $\sum\frac{1}{n_k}<\infty$ then $f(z)$ assumes every value at least once, and Biernacki \cite{Bi28} proved that if $\sum\frac{1}{n_k}<\infty$ then $f(z)$ assumes every value infinitely often.
P\'{o}lya \cite{Po29} proved that if $f$ has finit... | 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 |
2,155 | EP-520 | Erdős Problem #520 | Let $f$ be a Rademacher multiplicative function: a random $\{-1,0,1\}$-valued multiplicative function, where for each prime $p$ we independently choose $f(p)\in \{-1,1\}$ uniformly at random, and for square-free integers $n$ we extend $f(p_1\cdots p_r)=f(p_1)\cdots f(p_r)$ (and $f(n)=0$ if $n$ is not squarefree). Does ... | Note that if we drop the multiplicative assumption, and simply assign $f(m)=\pm 1$ at random, then this statement is true (with $c=\sqrt{2}$), the law of the iterated logarithm.
Wintner \cite{Wi44} proved that, almost surely, $ \sum_{m\leq N}f(m)\ll N^{1/2+o(1)}, $ and Erd\H{o}s improved the right-hand side to $N^{1/2}... | 1 | open | 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 still lists #520 as “open” on 2026-09-27. The August literature triage claimed a resolution, but the discussion explicitly flags that conclusion as unverified. That claim is not accepted as a full solution here.
**What remains.** Resolve the discrepancy between the earlier triage c... | [
{
"url": "https://www.erdosproblems.com/520",
"label": "Erdős Problems #520: open"
},
{
"url": "https://arxiv.org/abs/2607.29429",
"label": "Benjamin Durkan and Andrew Pearce-Crump, A sharp almost sure upper bound for partial sums of random multiplicative functions, arXiv:2607.29429 (2026)."
}... | 2026-09-27T00:00:00 | null | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27T00:00:00",
"kind": "needs_review",
"evidence": "maintained_source"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,156 | EP-521 | Erdős Problem #521 | Let $(\epsilon_k)_{k\geq 0}$ be independently uniformly chosen at random from $\{-1,1\}$. If $R_n$ counts the number of real roots of $f_n(z)=\sum_{0\leq k\leq n}\epsilon_k z^k$ then is it true that, almost surely, $ \lim_{n\to \infty}\frac{R_n}{\log n}=\frac{2}{\pi}? $ | Erd\H{o}s and Offord \cite{EO56} showed that the number of real roots of a random degree $n$ polynomial with $\pm 1$ coefficients is $(\frac{2}{\pi}+o(1))\log n$.
It is ambiguous in \cite{Er61} whether Erd\H{o}s intended the coefficients to be uniformly chosen from $\{-1,1\}$ or $\{0,1\}$. In the latter case, the const... | 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 |
2,157 | EP-522 | Erdős Problem #522 | Let $f(z)=\sum_{0\leq k\leq n} \epsilon_k z^k$ be a random polynomial, where $\epsilon_k\in \{-1,1\}$ independently uniformly at random for $0\leq k\leq n$.
Is it true that, if $R_n$ is the number of roots of $f(z)$ in $\{ z\in \mathbb{C} : \lvert z\rvert \leq 1\}$, then $ \frac{R_n}{n/2}\to 1 $ almost surely? | Random polynomials with independently identically distributed coefficients are sometimes called Kac polynomials - this problem considers the case of Rademacher coefficients, i.e. independent uniform $\pm 1$ values. Erd\H{o}s and Offord \cite{EO56} showed that the number of real roots of a random degree $n$ polynomial w... | 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 |
2,158 | EP-524 | Erdős Problem #524 | For any $t\in (0,1)$ let $t=\sum_{k=1}^\infty \epsilon_k(t)2^{-k}$ (where $\epsilon_k(t)\in \{0,1\}$). What is the correct order of magnitude (for almost all $t\in(0,1)$) for $ M_n(t)=\max_{x\in [-1,1]}\left\lvert \sum_{k\leq n}(-1)^{\epsilon_k(t)}x^k\right\rvert? $ | A problem of Salem and Zygmund \cite{SaZy54}. Chung showed that, for almost all $t$, there exist infinitely many $n$ such that $ M_n(t) \ll \left(\frac{n}{\log\log n}\right)^{1/2}. $ Erd\H{o}s (unpublished) showed that for almost all $t$ and every $\epsilon>0$ we have $\lim_{n\to \infty}M_n(t)/n^{1/2-\epsilon}=\infty$.... | 1 | open | 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 still lists #524 as “open” on 2026-09-27. The August literature triage claimed a resolution, but the discussion explicitly flags that conclusion as unverified. That claim is not accepted as a full solution here.
**What remains.** Resolve the discrepancy between the earlier triage c... | [
{
"url": "https://www.erdosproblems.com/524",
"label": "Erdős Problems #524: open"
},
{
"url": "https://arxiv.org/abs/2604.19294",
"label": "Brayden Letwin and Mehtaab Sawhney, On the maxima of Littlewood polynomials on [-1,1], arXiv:2604.19294 (2026)."
},
{
"url": "https://doi.org/10.10... | 2026-09-27T00:00:00 | null | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27T00:00:00",
"kind": "needs_review",
"evidence": "maintained_source"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,159 | EP-528 | Erdős Problem #528 | Let $f(n,k)$ count the number of self-avoiding walks of $n$ steps (beginning at the origin) in $\mathbb{Z}^k$ (i.e. those walks which do not intersect themselves). Determine $ C_k=\lim_{n\to\infty}f(n,k)^{1/n}. $ | The constant $C_k$ is sometimes known as the connective constant. Hammersley and Morton \cite{HM54} showed that this limit exists, and it is trivial that $k\leq C_k\leq 2k-1$.
Kesten \cite{Ke63} proved that $C_k=2k-1-1/2k+O(1/k^2)$, and more precise asymptotics are given by Clisby, Liang, and Slade \cite{CLS07}.
Conway... | 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 |
2,160 | EP-529 | Erdős Problem #529 | Let $d_k(n)$ be the expected distance from the origin after taking $n$ random steps from the origin in $\mathbb{Z}^k$ (conditional on no self intersections) - that is, a self-avoiding walk. Is it true that $ \lim_{n\to \infty}\frac{d_2(n)}{n^{1/2}}= \infty? $ Is it true that $ d_k(n)\ll n^{1/2} $ for $k\geq 3$? | Slade \cite{Sl87} proved that, for $k$ sufficiently large, $d_k(n)\sim Dn^{1/2}$ for some constant $D>0$ (independent of $k$). Hara and Slade (\cite{HaSl91} and \cite{HaSl92}) proved this for all $k\geq 5$.
For $k=2$ Duminil-Copin and Hammond \cite{DuHa13} have proved that $d_2(n)=o(n)$.
It is now conjectured that $d_k... | 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 |
2,161 | EP-530 | Erdős Problem #530 | Let $\ell(N)$ be maximal such that in any finite set $A\subset \mathbb{R}$ of size $N$ there exists a Sidon subset $S$ of size $\ell(N)$ (i.e. the only solutions to $a+b=c+d$ in $S$ are the trivial ones). Determine the order of $\ell(N)$. | In particular, is it true that $\ell(N)\sim N^{1/2}$?
Originally asked by Riddell \cite{Ri69}. Erd\H{o}s noted the bounds $ N^{1/3} \ll \ell(N) \leq (1+o(1))N^{1/2} $ (the upper bound following from the case $A=\{1,\ldots,N\}$). The lower bound was improved to $N^{1/2}\ll \ell(N)$ by Koml\'{o}s, Sulyok, and Szemer\'{e}... | 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 |
2,162 | EP-531 | Erdős Problem #531 | Let $F(k)$ be the minimal $N$ such that if we two-colour $\{1,\ldots,N\}$ there is a set $A$ of size $k$ such that all subset sums $\sum_{a\in S}a$ (for $\emptyset
eq S\subseteq A$) are monochromatic. Estimate $F(k)$. | The existence of $F(k)$ was established by Sanders and Folkman, and it also follows from Rado's theorem. It is commonly known as Folkman's theorem.
Erd\H{o}s and Spencer \cite{ErSp89} proved that $ F(k) \geq 2^{ck^2/\log k} $ for some constant $c>0$. Balogh, Eberhrad, Narayanan, Treglown, and Wagner \cite{BENTW17} have... | 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 |
2,163 | EP-533 | Erdős Problem #533 | Let $\delta>0$. If $n$ is sufficiently large and $G$ is a graph on $n$ vertices with no $K_5$ and at least $\delta n^2$ edges then $G$ contains a set of $\gg_\delta n$ vertices containing no triangle. | A problem of Erd\H{o}s, Hajnal, Simonovits, S\'{o}s, and Szemer\'{e}di, who could prove this is true for $\delta>1/16$, and could further prove it for $\delta>0$ if we replace $K_5$ with $K_4$.
They further observed that it fails for $\delta =1/4$ if we replace $K_5$ with $K_7$: by a construction of Erd\H{o}s and Roger... | 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 533 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/533",
"label": "Erdős Problems #533: 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 #533: disproved (Lean): https://www.erdosproblems.com/533
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 |
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