Central trinomial coefficients

An integer n > 3 is prime if and only if a(n) ≡ 1 pmodn^2. We have verified this for n up to 8 · 10^5, and proved that a(p) ≡ 1 pmodp^2 for any prime p > 3 (cf. A277640). - Zhi-Wei Sun, Nov 30 2016

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-2426,
  title        = {Central trinomial coefficients},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-2426}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

An integer $n > 3$ is prime if and only if $a (n) \equiv 1 (mod n^{2})$ . We have verified this for $n$ up to $8 \cdot 1 0^{5}$ , and proved that $a (p) \equiv 1 (mod p^{2})$ for any prime $p > 3$ (cf. A277640).

Zhi-Wei Sun, Nov 30 2016

Central trinomial coefficients: largest coefficient of $(1 + x + x^{2})^{n}$ , which is the coefficient of $x^{n}$ in the expansion of $(1 + x + x^{2})^{n}$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«2426».

theorem conjecture (n : ℕ) (hn : 3 < n) :
    n.Prime ↔ a n ≡ 1 [MOD n ^ 2]

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

A002426

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.