| import OAI.RingTheory.DirectFiniteness.PrimeCertificate |
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|
| namespace OAI |
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|
| noncomputable section |
| open scoped Classical |
| namespace KaplanskyCounterexample.FiniteGeometry |
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|
| def sourceQ : ℕ := 2^217 |
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| lemma sourceQ_ge : 2^200 ≤ sourceQ := by |
| exact Nat.pow_le_pow_right (by decide : 0 < 2) (by decide : 200 ≤ 217) |
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| lemma sourceQ_large : 4800 ≤ sourceQ := by |
| exact (by norm_num : 4800 ≤ (2:ℕ)^200).trans sourceQ_ge |
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| lemma sourceQ_pos : 0 < sourceQ := lt_of_lt_of_le (by decide : 0 < 4800) sourceQ_large |
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| 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) |
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| 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)) |
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| 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 _ _ |
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| local instance : DecidablePred (fun p : ℕ => p.Prime ∧ sourceQ^100 ∣ p+1) := Classical.decPred _ |
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| def sourcePrime : ℕ := Nat.find source_prime_exists |
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| def sourceD : ℕ := sourcePrime+1 |
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| 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 |
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| 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 |
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|
| lemma sourceD_q_dvd : sourceQ ∣ sourceD := |
| (dvd_pow_self sourceQ (by decide : 100 ≠ 0)).trans sourceD_divisible |
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| 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 |
|
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| end OAI |
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