Balanced walks on a 4-dimensional cubic lattice with neighbor steps

"It appears that for primes p ≥ 5, a(p) ≡ 1 pmodp^5." - _Peter Bala_, Dec 12 2021

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.

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Cite

@misc{cairn-oeis-212334,
  title        = {Balanced walks on a 4-dimensional cubic lattice with neighbor steps},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-212334}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-11}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

conjecture_1. "It appears that for primes $p \geq 5$ , $a (p) \equiv 1 (mod p^{5})$ ."

_Peter Bala_, Dec 12 2021

conjecture_2. "Conjecture: for $r \geq 2$ , and all primes $p \geq 5$ , $a (p^{r}) \equiv a (p^{r - 1}) (mod p^{3 r + 3})$ ."

_Peter Bala_, Oct 13 2022

The sequence $a (n)$ counts the words over a $4$ -ary alphabet (either empty or starting with the first letter) in which each letter occurs $n$ times and adjacent letters are equal or adjacent in the alphabet. Equivalently, $a (n) = k = 0 \sum n - 1 (k n) (k n - 1) (k n + k - 1)^{2}$ for $n \geq 1$ and $a (0) = 1$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«212334» (2 statements).

theorem conjecture_1 (p : ℕ) (hp : p.Prime) (hp5 : 5 ≤ p) :
    a p % (p ^ 5) = 1

theorem conjecture_2 (p r : ℕ) (hp : p.Prime) (hp5 : 5 ≤ p) (hr2 : 2 ≤ r) :
    a (p ^ r) % (p ^ (3 * r + 3)) = a (p ^ (r - 1)) % (p ^ (3 * r + 3))

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
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

A212334

Source and licence

Imported from Formal Conjectures (OEIS), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.