363102.lean`

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

8 declarations, 1 contain sorry, 0 declared as axioms

Primality of continued fraction denominators $a(n)$ for $n \ge 3$

Auxiliary sequence A051403, defined as $$\frac{(n+2) \sum_{k=0}^n k!}{2}$$

Denominator of the continued fraction $1/(2-3/(3-4/(4-5/(...(n-1)-n/(-2)))))$. The sequence is defined by the formula: $$a(n) = \frac{n^2 - 2}{\gcd(n^2 - 2, 2 \cdot A051403(n-3) + n \cdot A051403(n-4))}$$ The formula is valid for $n \ge 3$.

References:

A363102
arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.
A051403

Imports (1)

FormalConjecturesUtil

def OeisA363102.a051403 Auxiliary sequence A051403, defined as $$\frac{(n+2) \sum_{k=0}^n k!}{2}$$ line 48 source
Source documentation
Auxiliary sequence A051403, defined as $$\frac{(n+2) \sum_{k=0}^n k!}{2}$$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.sum, Finset.range, HAdd.hAdd, OfNat.ofNat, Nat.factorial, HDiv.hDiv, HMul.hMul
8 instances, coercions and auxiliary constants left out.
```
def a051403 (n : ℕ) : ℕ :=
…
```
def OeisA363102.a Denominator of the continued fraction $1/(2-3/(3-4/(4-5/(...(n-1)-n/(-2)))))$ line 58 source
Source documentation
Denominator of the continued fraction $1/(2-3/(3-4/(4-5/(...(n-1)-n/(-2)))))$. The sequence is defined by the formula: $$a(n) = \frac{n^2 - 2}{\gcd(n^2 - 2, 2 \cdot A051403(n-3) + n \cdot A051403(n-4))}$$ The formula is valid for $n \ge 3$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: HSub.hSub, HPow.hPow, OfNat.ofNat, OeisA363102.a051403, HAdd.hAdd, HMul.hMul, HDiv.hDiv, Nat.gcd
13 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
lemma OeisA363102.a_3 line 68 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA363102.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 OeisA363102.a_4 line 71 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA363102.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 OeisA363102.a_5 line 74 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA363102.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 OeisA363102.a_6 line 77 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA363102.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 OeisA363102.a_7 line 80 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA363102.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 OeisA363102.a_eq_one_or_prime sorry line 91 source
depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Or, Eq, OeisA363102.a, Nat.Prime
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

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