A binomial coefficient sum

Let b(n) = a(2n-1). Then the supercongruence b(n p^k) ≡ b(n p^k-1) pmodp^3k holds for positive integers n and k and all primes p ≥ 5. - Zhi-Wei Sun, Nov 16 2019

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-oeis-3161,
  title        = {A binomial coefficient sum},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-3161}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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On the literature board

The problem

The question

Let $b (n) = a (2 n - 1)$ . Then the supercongruence $b (n p^{k}) \equiv b (n p^{k - 1}) (mod p^{3 k})$ holds for positive integers $n$ and $k$ and all primes $p \geq 5$ .

Zhi-Wei Sun, Nov 16 2019

A binomial coefficient sum: $a (n) = k = 0 \sum ⌊ n /2 ⌋ ((k n) - (k - 1 n))^{3}$ where $(- 1 n) = 0$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«3161».

theorem conjecture (n k p : ℕ) (hn : 0 < n) (hk : 0 < k) (hp : p.Prime) (hp_ge : 5 ≤ p) :
    (b (n * p ^ k) : ℤ) ≡ (b (n * p ^ (k - 1)) : ℤ) [ZMOD (p : ℤ) ^ (3 * k)]

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

A003161

Source and licence

Imported from Formal Conjectures (OEIS), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.