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import FormalConjecturesUtilSmallest prime factors of $2^{2^n+2} + 3$
Smallest prime factor of $2^{2^n+2} + 3$.
References:
arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.
namespace OeisA248802Smallest prime factor of $2^{2^n+2} + 3$.
def a (n : ℕ) : ℕ := (2 ^ (2 ^ n + 2) + 3).minFacAn index k is covered by Conjecture 1 if k = 10m + 2 for some m >= 0, predicting a(k)=67.
def CoveredByC1 (k : ℕ) : Prop := ∃ m : ℕ, k = 10 * m + 2An index k is covered by Conjecture 2 if k = 36m + 16 for some m >= 0, and m is not 1 mod 5, predicting a(k)=271.
def CoveredByC2 (k : ℕ) : Prop := ∃ m : ℕ, k = 36 * m + 16 ∧ m % 5 ≠ 1An index k is covered by Conjecture 3 if k = 84m + 22 for some m >= 0, and m is not 0 mod 5, predicting a(k)=523.
def CoveredByC3 (k : ℕ) : Prop := ∃ m : ℕ, k = 84 * m + 22 ∧ m % 5 ≠ 0@[category test, AMS 11]
lemma a_0 : a 0 = 11 := ⊢ a 0 = 11 All goals completed! 🐙@[category test, AMS 11]
lemma a_1 : a 1 = 19 := ⊢ a 1 = 19 All goals completed! 🐙@[category test, AMS 11]
lemma a_2 : a 2 = 67 := ⊢ a 2 = 67 All goals completed! 🐙@[category test, AMS 11]
lemma a_3 : a 3 = 13 := ⊢ a 3 = 13 All goals completed! 🐙@[category test, AMS 11]
lemma a_4 : a 4 = 262147 := ⊢ a 4 = 262147 All goals completed! 🐙Conjecture 1: $a(10n+2) = 67$ for $n \ge 0$. - Chai Wah Wu, Oct 21 2019
A formal proof has been found with the methods described in arxiv/2605.22763.
@[category research solved, AMS 11, formal_proof using formal_conjectures at
"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/248802.wip.lean#L208"]
theorem a_ten_mul_add_two_eq (n : ℕ) : a (10 * n + 2) = 67 := n:ℕ⊢ a (10 * n + 2) = 67
All goals completed! 🐙Conjecture 4: $a(58n+26) = 1399$ for $n \ge 0$ and when it is not covered by Conjectures 1-3. - Chai Wah Wu, Oct 21 2019
A formal proof has been found with the methods described in arxiv/2605.22763.
@[category research solved, AMS 11, formal_proof using formal_conjectures at
"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/248802.wip.lean#L982"]
theorem a_fifty_eight_mul_add_twenty_six_eq (n : ℕ) :
(¬CoveredByC1 (58 * n + 26) ∧ ¬CoveredByC2 (58 * n + 26) ∧
¬CoveredByC3 (58 * n + 26)) → a (58 * n + 26) = 1399 := n:ℕ⊢ ¬CoveredByC1 (58 * n + 26) ∧ ¬CoveredByC2 (58 * n + 26) ∧ ¬CoveredByC3 (58 * n + 26) →
OeisA248802.a (58 * n + 26) = 1399
All goals completed! 🐙end OeisA248802