5258.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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

8 declarations, 1 contain sorry, 0 declared as axioms

Apéry numbers

Apéry numbers: $$a(n) = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}$$

References:

A005258

Imports (1)

FormalConjecturesUtil

def OeisA5258.a Apéry numbers: $a(n) = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}$ line 35 source
Source documentation
Apéry numbers: $a(n) = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.sum, Finset.range, HAdd.hAdd, OfNat.ofNat, HMul.hMul, HPow.hPow, Nat.choose
10 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA5258.a_0 line 39 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA5258.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 OeisA5258.a_1 line 42 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA5258.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 OeisA5258.a_2 line 45 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA5258.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 OeisA5258.a_3 line 48 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA5258.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 OeisA5258.a_4 line 51 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA5258.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.
def OeisA5258.aperyPoly The polynomial associated with the $n$-th Apéry number: $a_n(x) = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k} x^k$ line 56 source
Source documentation
The polynomial associated with the $n$-th Apéry number: $a_n(x) = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k} x^k$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Polynomial, Rat
Proof uses: Finset.sum, Finset.range, HAdd.hAdd, OfNat.ofNat, HMul.hMul, RingHom, Polynomial.C, HPow.hPow, Nat.choose, Polynomial.X
20 instances, coercions and auxiliary constants left out.
```
noncomputable def aperyPoly (n : ℕ) : ℚ[X] :=
…
```
theorem OeisA5258.conjecture For each $n = 1, 2, 3, \dots$ the polynomial $a_n(x) = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k} x^k$ is irreducible over the field of rational numbers sorry line 67 source
Source documentation
For each $n = 1, 2, 3, \dots$ the polynomial $a_n(x) = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k} x^k$ is irreducible over the field of rational numbers.
- Zhi-Wei Sun, Mar 21 2013
depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Irreducible, Polynomial, Rat, OeisA5258.aperyPoly
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.
```
theorem conjecture (n : ℕ) (hn : 1 ≤ n) : Irreducible (aperyPoly n) := by
…
```
Mentions: OeisA5258.aperyPoly