357513.lean`

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L4DEL: Lean4 Domain Explanation Language

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

11 declarations, 2 contain sorry, 0 declared as axioms

Numerator of a sum involving binomial coefficients

$a(n)$ is the numerator of $\sum_{k = 1}^n \frac{1}{k^3} \binom{n}{k}^2 \binom{n+k}{k}^2$ for $n \ge 1$ with $a(0) = 0$.

References:

A357513

Imports (1)

FormalConjecturesUtil

def OeisA357513.a A357513: $a(n) = \text{numerator of } \sum_{k = 1..n} \frac{1}{k^3} \binom{n}{k}^2 \binom{n+k}{k}^2 \text{ for } n \ge 1 \text{ with } a(0) = 0$ line 41 source
Source documentation
A357513: $a(n) = \text{numerator of } \sum_{k = 1..n} \frac{1}{k^3} \binom{n}{k}^2 \binom{n+k}{k}^2 \text{ for } n \ge 1 \text{ with } a(0) = 0$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Int.natAbs, Rat.num, Finset.sum, Rat, Finset.Icc, OfNat.ofNat, HDiv.hDiv, HMul.hMul, HPow.hPow, Nat.choose, HAdd.hAdd
16 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA357513.a_0 line 45 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA357513.a, OfNat.ofNat
Proof uses: rfl
2 instances, coercions and auxiliary constants left out.
theorem OeisA357513.a_1 line 48 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA357513.a, OfNat.ofNat
Proof uses: id, of_eq_true, Int.natAbs, Rat.num, Finset.sum, Rat, Finset.Icc, HDiv.hDiv, HMul.hMul, HPow.hPow, Nat.choose, HAdd.hAdd, Eq.trans, True, congrFun', congrArg, Int, Singleton.singleton, Finset, Finset.sum_congr and 33 more
46 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA357513.a_2 line 53 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA357513.a, OfNat.ofNat
Proof uses: id, of_eq_true, Int.natAbs, Rat.num, Finset.sum, Rat, Finset.Icc, HDiv.hDiv, HMul.hMul, HPow.hPow, Nat.choose, HAdd.hAdd, Eq.trans, True, congrFun', congrArg, Int, Finset.sum_Icc_succ_top, Mathlib.Meta.NormNum.isNat_le_true, Mathlib.Meta.NormNum.isNat_ofNat and 42 more
58 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA357513.a_3 line 58 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA357513.a, OfNat.ofNat
Proof uses: id, of_eq_true, Int.natAbs, Rat.num, Finset.sum, Rat, Finset.Icc, HDiv.hDiv, HMul.hMul, HPow.hPow, Nat.choose, HAdd.hAdd, Eq.trans, True, congrFun', congrArg, Int, Finset.sum_Icc_succ_top, Mathlib.Meta.NormNum.isNat_le_true, Mathlib.Meta.NormNum.isNat_ofNat and 42 more
58 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA357513.a_4 line 63 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA357513.a, OfNat.ofNat
Proof uses: id, of_eq_true, Int.natAbs, Rat.num, Finset.sum, Rat, Finset.Icc, HDiv.hDiv, HMul.hMul, HPow.hPow, Nat.choose, HAdd.hAdd, Eq.trans, True, congrFun', congrArg, Int, Finset.sum_Icc_succ_top, Mathlib.Meta.NormNum.isNat_le_true, Mathlib.Meta.NormNum.isNat_ofNat and 42 more
58 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA357513.a_5 line 68 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA357513.a, OfNat.ofNat
Proof uses: id, of_eq_true, Int.natAbs, Rat.num, Finset.sum, Rat, Finset.Icc, HDiv.hDiv, HMul.hMul, HPow.hPow, Nat.choose, HAdd.hAdd, Eq.trans, True, congrFun', congrArg, Int, Finset.sum_Icc_succ_top, Mathlib.Meta.NormNum.isNat_le_true, Mathlib.Meta.NormNum.isNat_ofNat and 42 more
58 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA357513.a357513_supercongruence sorry line 80 source
depends on sorry what this means
Statement uses: Nat, Nat.Prime, GE.ge, OfNat.ofNat, Ne, Int.ModEq, HPow.hPow, Int, OeisA357513.a, HSub.hSub
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
def OeisA357513.u Let m be a nonnegative integer and set line 90 source
Source documentation
Let m be a nonnegative integer and set $u(n)$ = the numerator of $$\sum{k=1}^{n} \frac{1}{k^{2m+1}} \binom{n}{k}^2 \binom{n+k}{k}^2$$ (seems like a typo in the OEIS entry: the sum starts with $k=0$ there. In order to avoid a division by zero, we replace start the sum at $k=1$.)

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Int.natAbs, Rat.num, Finset.sum, Rat, Finset.Icc, OfNat.ofNat, HDiv.hDiv, HMul.hMul, HPow.hPow, Nat.choose, HAdd.hAdd
18 instances, coercions and auxiliary constants left out.
```
noncomputable def u (m : ℕ) (n : ℕ) : ℕ :=
…
```
theorem OeisA357513.general_supercongruence We conjecture that $u(p-1) == 0 (mod p^4)$ for all primes $p$, with a finite number of exceptions that depend on $m$ sorry line 99 source
Source documentation
We conjecture that $u(p-1) == 0 (mod p^4)$ for all primes $p$, with a finite number of exceptions that depend on $m$.

depends on sorry what this means
Statement uses: Nat, Exists, Finset, Nat.Prime, Not, Membership.mem, Eq, ZMod, HPow.hPow, OfNat.ofNat, OeisA357513.u, HSub.hSub
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem general_supercongruence (m : ℕ) : ∃ (exceptions : Finset ℕ), ∀ p, p.Prime →
    p ∉ exceptions → u m (p - 1) = (0 : ZMod (p ^ 4)) := by
…
```
Mentions: OeisA357513.u
theorem OeisA357513.general_supercongruence_one_of_a357513_supercongruence line 104 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Nat.Prime, GE.ge, OfNat.ofNat, Ne, Int.ModEq, HPow.hPow, Int, OeisA357513.a, HSub.hSub, Exists, Finset, Not, Membership.mem, Eq, ZMod, OeisA357513.u
Proof uses: Exists.intro, Insert.insert, Singleton.singleton, Eq.mpr, propext, id, Iff, Int.natAbs, Rat.num, Finset.sum, Rat, Finset.Icc, HDiv.hDiv, HMul.hMul, Nat.choose, HAdd.hAdd, of_eq_true, Eq.trans, Dvd.dvd, abs and 25 more
83 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.