79727.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

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

$a(n) = \sum_{k=0}^n \binom{2k}{k}^3$

The sequence $a(n) = 1 + \binom{2}{1}^3 + \binom{4}{2}^3 + \cdots + \binom{2n}{n}^3$: $$a(n) = \sum_{k=0}^n \binom{2k}{k}^3$$

References:

A079727

Imports (2)

def OeisA79727.a $a(n) = \sum_{k=0}^n \binom{2k}{k}^3$ line 35 source
Source documentation
$a(n) = \sum_{k=0}^n \binom{2k}{k}^3$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.sum, Finset.range, HAdd.hAdd, OfNat.ofNat, HPow.hPow, Nat.choose, HMul.hMul
10 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA79727.a_0 Value of the sequence a at 0 line 40 source
Source documentation
Value of the sequence a at 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA79727.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 a_0 : a 0 = 1 := by
…
```
Mentions: OeisA79727.a
theorem OeisA79727.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, Nat, OeisA79727.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 a_1 : a 1 = 9 := by
…
```
Mentions: OeisA79727.a
theorem OeisA79727.a_2 Value of the sequence a at 2 line 50 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA79727.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 a_2 : a 2 = 225 := by
…
```
Mentions: OeisA79727.a
theorem OeisA79727.a_3 Value of the sequence a at 3 line 55 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA79727.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 a_3 : a 3 = 8225 := by
…
```
Mentions: OeisA79727.a
theorem OeisA79727.a_4 Value of the sequence a at 4 line 60 source
Source documentation
Value of the sequence a at 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA79727.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 a_4 : a 4 = 351225 := by
…
```
Mentions: OeisA79727.a
theorem OeisA79727.conjecture1 Conjecture 1 (Peter Bala, 2024): If prime $p$ is in A003625 then $a(p^2) \equiv 8 + p^2 \pmod{p^3}$ sorry line 68 source
Source documentation
Conjecture 1 (Peter Bala, 2024): If prime $p$ is in A003625 then $a(p^2) \equiv 8 + p^2 \pmod{p^3}$.

depends on sorry what this means
Statement uses: Nat, OeisA3625.A, Nat.ModEq, HPow.hPow, OfNat.ofNat, OeisA79727.a, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture1 (p : ℕ) (hp : OeisA3625.A p) :
    a (p ^ 2) ≡ 8 + p ^ 2 [MOD p ^ 3] := by
…
```
Mentions: OeisA3625.A, OeisA79727.a
theorem OeisA79727.conjecture2 Conjecture 2 (Peter Bala, 2024): If prime $p$ is in A003625 then $a(p(p-1)) \equiv p^2 \pmod{p^3}$ sorry line 77 source
Source documentation
Conjecture 2 (Peter Bala, 2024): If prime $p$ is in A003625 then $a(p(p-1)) \equiv p^2 \pmod{p^3}$.

depends on sorry what this means
Statement uses: Nat, OeisA3625.A, Nat.ModEq, HPow.hPow, OfNat.ofNat, OeisA79727.a, HMul.hMul, HSub.hSub
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture2 (p : ℕ) (hp : OeisA3625.A p) :
    a (p * (p - 1)) ≡ p ^ 2 [MOD p ^ 3] := by
…
```
Mentions: OeisA3625.A, OeisA79727.a
theorem OeisA79727.conjecture3 Conjecture 3 (Peter Bala, 2024): If prime $p$ is in A003625 then $a((p^2-1)/2) \equiv p^2 \pmod{p^4}$ sorry line 86 source
Source documentation
Conjecture 3 (Peter Bala, 2024): If prime $p$ is in A003625 then $a((p^2-1)/2) \equiv p^2 \pmod{p^4}$.

depends on sorry what this means
Statement uses: Nat, OeisA3625.A, Nat.ModEq, HPow.hPow, OfNat.ofNat, OeisA79727.a, HDiv.hDiv, HSub.hSub
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture3 (p : ℕ) (hp : OeisA3625.A p) :
    a ((p ^ 2 - 1) / 2) ≡ p ^ 2 [MOD p ^ 4] := by
…
```
Mentions: OeisA3625.A, OeisA79727.a
theorem OeisA79727.conjecture4 Conjecture 4 (Peter Bala, 2024): If $n$ is a product of distinct primes from A003625 then $a((n-1)/2)$ is divisible by $n^2$ sorry line 95 source
Source documentation
Conjecture 4 (Peter Bala, 2024): If $n$ is a product of distinct primes from A003625 then $a((n-1)/2)$ is divisible by $n^2$.

depends on sorry what this means
Statement uses: Finset, Nat, Membership.mem, OeisA3625.A, Dvd.dvd, HPow.hPow, Finset.prod, OfNat.ofNat, OeisA79727.a, HDiv.hDiv, HSub.hSub
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: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture4 (S : Finset ℕ) (hS : ∀ p ∈ S, OeisA3625.A p) :
    (∏ p ∈ S, p) ^ 2 ∣ a ((∏ p ∈ S, p - 1) / 2) := by
…
```
Mentions: OeisA3625.A, OeisA79727.a