307865.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

8 declarations, 1 contain sorry, 0 declared as axioms

Vanishing 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:

A307865
arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.

Imports (1)

FormalConjecturesUtil

def OeisA307865.a $a(n)$ is the number of natural bases $b < 2n+1$ such that $b^n \equiv -1 \pmod{2n+1}$ line 44 source
Source documentation
$a(n)$ 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 $\mathbb{Z}/(2n+1)\mathbb{Z}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: HAdd.hAdd, HMul.hMul, OfNat.ofNat, Finset.card, Finset.filter, Eq, ZMod, HPow.hPow, Neg.neg, Finset.Ico
28 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
def OeisA307865.IsAbsoluteEulerPseudoprime A natural number $m > 1$ is an absolute Euler pseudoprime if it is composite and for all $b$ coprime to $m$, $b^{(m-1)/2} \equiv \pm 1 \pmod m$ line 55 source
Source documentation
A natural number $m > 1$ is an absolute Euler pseudoprime if it is composite and for all $b$ coprime to $m$, $b^{(m-1)/2} \equiv \pm 1 \pmod m$.

standard axioms: Quot.sound, propext what this means
Statement uses: Nat
Proof uses: And, GT.gt, OfNat.ofNat, Not, Nat.Prime, Nat.Coprime, Or, Eq, ZMod, HPow.hPow, HDiv.hDiv, HSub.hSub, Neg.neg
26 instances, coercions and auxiliary constants left out.
```
def IsAbsoluteEulerPseudoprime (m : ℕ) : Prop :=
…
```
lemma OeisA307865.a_0 line 61 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA307865.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma OeisA307865.a_1 line 64 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA307865.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma OeisA307865.a_2 line 67 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA307865.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma OeisA307865.a_3 line 70 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA307865.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma OeisA307865.a_4 line 73 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA307865.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA307865.a_eq_zero_of_pseudoprime sorry line 84 source
depends on sorry what this means
Statement uses: Nat, OeisA307865.IsAbsoluteEulerPseudoprime, HAdd.hAdd, HMul.hMul, OfNat.ofNat, Eq, OeisA307865.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.