179537.lean`

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L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

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

10 declarations, 4 contain sorry, 0 declared as axioms

Central binomial sum $a(n) = \sum_{k=0}^n \binom{n}{k}^2 \binom{n-k}{k}^2 (-16)^k$

The sequence is defined by $$a(n) = \sum_{k=0}^n \binom{n}{k}^2 \binom{n-k}{k}^2 (-16)^k.$$

References:

A179537
Z.-W. Sun, "Open Conjectures on Congruences", arXiv preprint arXiv:0911.5665 [math.NT], 2009-2011.

Imports (1)

FormalConjecturesUtil

def OeisA179537.a The sequence $a(n) = \sum_{k=0}^n \binom{n}{k}^2 \binom{n-k}{k}^2 (-16)^k$ line 36 source
Source documentation
The sequence $a(n) = \sum_{k=0}^n \binom{n}{k}^2 \binom{n-k}{k}^2 (-16)^k$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Int
Proof uses: Finset.sum, Finset.range, HAdd.hAdd, OfNat.ofNat, HMul.hMul, HPow.hPow, Nat.choose, HSub.hSub, Neg.neg
16 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℤ :=
…
```
theorem OeisA179537.a_0 Value of the sequence a at 0 line 41 source
Source documentation
Value of the sequence a at 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA179537.a, OfNat.ofNat, Nat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_0 : a 0 = 1 := by decide
…
```
Mentions: OeisA179537.a
theorem OeisA179537.a_1 Value of the sequence a at 1 line 45 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA179537.a, OfNat.ofNat, Nat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = 1 := by decide
…
```
Mentions: OeisA179537.a
theorem OeisA179537.a_2 Value of the sequence a at 2 line 49 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA179537.a, OfNat.ofNat, Nat, Neg.neg
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
7 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : a 2 = -63 := by decide
…
```
Mentions: OeisA179537.a
theorem OeisA179537.a_3 Value of the sequence a at 3 line 53 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA179537.a, OfNat.ofNat, Nat, Neg.neg
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
7 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_3 : a 3 = -575 := by decide
…
```
Mentions: OeisA179537.a
theorem OeisA179537.a_4 Value of the sequence a at 4 line 57 source
Source documentation
Value of the sequence a at 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA179537.a, OfNat.ofNat, Nat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_4 : a 4 = 6913 := by decide
…
```
Mentions: OeisA179537.a
theorem OeisA179537.conjecture1 If $p$ is a prime with $(p/7) = 1$ and $p = x^2 + 7y^2$ with $x, y$ integers, then $\sum_{k=0}^{p-1} (-1)^k a(k) \equiv 4x^2 - 2p \pmod{p^2}$ sorry line 65 source
Source documentation
If $p$ is a prime with $(p/7) = 1$ and $p = x^2 + 7y^2$ with $x, y$ integers, then $\sum_{k=0}^{p-1} (-1)^k a(k) \equiv 4x^2 - 2p \pmod{p^2}$.
- Zhi-Wei Sun, Jul 17 2010
depends on sorry what this means
Statement uses: Nat, Fact, Nat.Prime, Ne, OfNat.ofNat, Eq, Int, legendreSym, Fact.mk, HAdd.hAdd, HPow.hPow, HMul.hMul, Int.ModEq, Finset.sum, Finset.range, Neg.neg, OeisA179537.a, HSub.hSub
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture1 (p : ℕ) [Fact p.Prime] (hp7 : p ≠ 7)
    (h_leg : letI : Fact (Nat.Prime 7) := ⟨by decide⟩; legendreSym 7 p = 1)
…
```
theorem OeisA179537.conjecture2 If $p$ is a prime with $(p/7) = -1$, then $\sum_{k=0}^{p-1} (-1)^k a(k) \equiv 0 \pmod{p^2}$ sorry line 78 source
Source documentation
If $p$ is a prime with $(p/7) = -1$, then $\sum_{k=0}^{p-1} (-1)^k a(k) \equiv 0 \pmod{p^2}$.
- Zhi-Wei Sun, Jul 17 2010
depends on sorry what this means
Statement uses: Nat, Fact, Nat.Prime, Ne, OfNat.ofNat, Eq, Int, legendreSym, Fact.mk, Neg.neg, Int.ModEq, HPow.hPow, Finset.sum, Finset.range, HMul.hMul, OeisA179537.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture2 (p : ℕ) [Fact p.Prime] (hp7 : p ≠ 7)
    (h_leg : letI : Fact (Nat.Prime 7) := ⟨by decide⟩; legendreSym 7 p = -1) :
…
```
theorem OeisA179537.conjecture3 $\sum_{k=0}^{n-1} (42k + 37) (-1)^k a(k) \equiv 0 \pmod n$ for all $n \ge 1$ sorry line 88 source
Source documentation
$\sum_{k=0}^{n-1} (42k + 37) (-1)^k a(k) \equiv 0 \pmod n$ for all $n \ge 1$.
- Zhi-Wei Sun, Jul 17 2010
depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Dvd.dvd, Int, Finset.sum, Finset.range, HMul.hMul, HAdd.hAdd, HPow.hPow, Neg.neg, OeisA179537.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
32 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture3 (n : ℕ) (hn : 1 ≤ n) :
    (n : ℤ) ∣ ∑ k ∈ Finset.range n, ((42 * (k : ℤ) + 37) * (-1 : ℤ) ^ k * a k) := by
…
```
Mentions: OeisA179537.a
theorem OeisA179537.conjecture4 $\sum_{k=0}^{p-1} (42k + 37) (-1)^k a(k) \equiv p(21(p/7) + 16) \pmod{p^2}$ for any prime $p \ne 7$ sorry line 97 source
Source documentation
$\sum_{k=0}^{p-1} (42k + 37) (-1)^k a(k) \equiv p(21(p/7) + 16) \pmod{p^2}$ for any prime $p \ne 7$.
- Zhi-Wei Sun, Jul 17 2010
depends on sorry what this means
Statement uses: Nat, Fact, Nat.Prime, Ne, OfNat.ofNat, Int.ModEq, HPow.hPow, Int, Finset.sum, Finset.range, HMul.hMul, HAdd.hAdd, Neg.neg, OeisA179537.a, legendreSym, Fact.mk
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
30 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture4 (p : ℕ) [Fact p.Prime] (_hp7 : p ≠ 7) :
    letI : Fact (Nat.Prime 7) := ⟨by decide⟩
…
```