File size: 3,147 Bytes
e3e0a0d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 | import OAI.RingTheory.DirectFiniteness.PrimeCertificate
namespace OAI
noncomputable section
open scoped Classical
namespace KaplanskyCounterexample.FiniteGeometry
def sourceQ : ℕ := 2^217
lemma sourceQ_ge : 2^200 ≤ sourceQ := by
exact Nat.pow_le_pow_right (by decide : 0 < 2) (by decide : 200 ≤ 217)
lemma sourceQ_large : 4800 ≤ sourceQ := by
exact (by norm_num : 4800 ≤ (2:ℕ)^200).trans sourceQ_ge
lemma sourceQ_pos : 0 < sourceQ := lt_of_lt_of_le (by decide : 0 < 4800) sourceQ_large
lemma exponent_comparison (x : ℕ) (hx : 0 < x) : (x^13)^1200 ≤ (x^217)^72 := by
rw [← pow_mul,← pow_mul]
exact pow_le_pow_right₀ (by omega : 1 ≤ x) (by decide : 13*1200 ≤ 217*72)
lemma sourceQ_density : (4800:ℕ)^1200 ≤ sourceQ^72 := by
exact (pow_le_pow_left' (by norm_num : (4800:ℕ) ≤ 2^13) 1200).trans
(exponent_comparison 2 (by decide))
lemma source_prime_exists : ∃ p : ℕ, p.Prime ∧ sourceQ^100 ∣ p+1 := by
have hp : 2*sourceQ^100-1 = mersenne 21701 := by
unfold sourceQ mersenne
have h : 21701=217*100+1 := by decide
rw [h,pow_add,pow_mul,pow_one]
rw [Nat.mul_comm]
refine ⟨2*sourceQ^100-1,(by rw [hp]; exact mersenne_21701_prime),?_⟩
have hpos : 0 < sourceQ^100 := pow_pos sourceQ_pos _
rw [Nat.sub_add_cancel (by omega : 1 ≤ 2*sourceQ^100)]
exact dvd_mul_left _ _
local instance : DecidablePred (fun p : ℕ => p.Prime ∧ sourceQ^100 ∣ p+1) := Classical.decPred _
def sourcePrime : ℕ := Nat.find source_prime_exists
def sourceD : ℕ := sourcePrime+1
lemma sourcePrime_prime : sourcePrime.Prime := (Nat.find_spec source_prime_exists).1
lemma sourcePrime_le (p : ℕ) (hp : p.Prime) (hc : sourceQ^100 ∣ p+1) : sourcePrime ≤ p :=
Nat.find_min' source_prime_exists ⟨hp,hc⟩
lemma sourceD_divisible : sourceQ^100 ∣ sourceD := (Nat.find_spec source_prime_exists).2
lemma sourceD_bounds : sourceQ^100 ≤ sourceD ∧ sourceD ≤ 2*sourceQ^100 := by
refine ⟨Nat.le_of_dvd (by unfold sourceD; omega) sourceD_divisible,?_⟩
have hp : 2*sourceQ^100-1 = mersenne 21701 := by
unfold sourceQ mersenne
have h : 21701=217*100+1 := by decide
rw [h,pow_add,pow_mul,pow_one,Nat.mul_comm]
have hd : sourceQ^100 ∣ (2*sourceQ^100-1)+1 := by
rw [Nat.sub_add_cancel (by have := pow_pos sourceQ_pos 100; omega)]
exact dvd_mul_left _ _
have hb : sourcePrime ≤ 2*sourceQ^100-1 := Nat.find_min' source_prime_exists ⟨(by rw [hp]; exact mersenne_21701_prime),hd⟩
unfold sourceD
have := pow_pos sourceQ_pos 100
omega
lemma sourceD_q_dvd : sourceQ ∣ sourceD :=
(dvd_pow_self sourceQ (by decide : 100 ≠ 0)).trans sourceD_divisible
lemma sourceD_log : Nat.log sourceQ sourceD = 100 := by
apply Nat.log_eq_of_pow_le_of_lt_pow
· exact sourceD_bounds.1
· calc
sourceD ≤ 2*sourceQ^100 := sourceD_bounds.2
_ < sourceQ^100*sourceQ := by
rw [Nat.mul_comm 2]
exact Nat.mul_lt_mul_of_pos_left (lt_of_lt_of_le (by decide : 2 < 4800) sourceQ_large)
(pow_pos sourceQ_pos _)
_ = sourceQ^(100+1) := (pow_succ _ _).symm
end KaplanskyCounterexample.FiniteGeometry
end
end OAI
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