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import FormalConjecturesUtilVanishing of bases with $b^n \equiv -1 \pmod{2n+1}$ for absolute Euler pseudoprimes
$a(n)$ is the number of natural bases $b < 2n+1$ such that $b^n \equiv -1 \pmod{2n+1}$.
If $2n+1$ is an absolute Euler pseudoprime, then $a(n) = 0$.
References:
arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.
namespace OeisA307865open Nat Finset ZMod$a(n)$ is the number of natural bases $b < 2n+1$ such that $b^n \equiv -1 \pmod{2n+1}$. The bases $b$ are interpreted as $b \in {1, 2, \dots, 2n}$. We check the condition in the ring $\mathbb{Z}/(2n+1)\mathbb{Z}$.
def a (n : ℕ) : ℕ :=
let m : ℕ := 2 * n + 1
-- The set of bases is $\{1, 2, \dots, 2n\} = \text{Ico } 1 m$.
(Ico 1 m).filter (fun b : ℕ => (b : ZMod m) ^ n = (-1 : ZMod m)) |>.cardvariable {n : ℕ}A natural number $m > 1$ is an absolute Euler pseudoprime if it is composite and for all $b$ coprime to $m$, $b^{(m-1)/2} \equiv \pm 1 \pmod m$.
def IsAbsoluteEulerPseudoprime (m : ℕ) : Prop :=
m > 1 ∧ ¬ Nat.Prime m ∧
(∀ b : ℕ, Nat.Coprime b m → (b : ZMod m) ^ ((m - 1) / 2) = 1 ∨ (b : ZMod m) ^ ((m - 1) / 2) = -1)@[category test, AMS 11]
lemma a_0 : a 0 = 0 := ⊢ a 0 = 0 All goals completed! 🐙@[category test, AMS 11]
lemma a_1 : a 1 = 1 := ⊢ a 1 = 1 All goals completed! 🐙@[category test, AMS 11]
lemma a_2 : a 2 = 2 := ⊢ a 2 = 2 All goals completed! 🐙@[category test, AMS 11]
lemma a_3 : a 3 = 3 := ⊢ a 3 = 3 All goals completed! 🐙@[category test, AMS 11]
lemma a_4 : a 4 = 0 := ⊢ a 4 = 0 All goals completed! 🐙Conjecture: if $2n+1$ is an absolute Euler pseudoprime, then $a(n) = 0$.
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/307865.wip.lean#L190"]
theorem a_eq_zero_of_pseudoprime (h : IsAbsoluteEulerPseudoprime (2 * n + 1)) : a n = 0 := n:ℕh:IsAbsoluteEulerPseudoprime (2 * n + 1)⊢ a n = 0
All goals completed! 🐙end OeisA307865