bibkey: oeis2024a375178 authors: Peter Bala year: 2024 title: A375178 — binomial cube sums and a fifth-power supercongruence doi: null url: https://oeis.org/A375178 claim: For every prime p at least 7, the entry conjectures that its binomial cube sum is 1 modulo p^5. strata_touched:

• D5/S3/ArithSums/A375178Supercongruence license: citation-only triage: anchor

The fifth-power conjecture in A375178

Verified locator

doi: null url: https://oeis.org/A375178

The internal entry https://oeis.org/A375178/internal defines a(n) as the sum of choose(n+k-1,k)^3 for 0 <= k < n, with offset zero, and explicitly labels the p^5 statement a conjecture. Its only bibliographical link is Meštrović’s 2011 survey, https://arxiv.org/abs/1111.3057.

The related entry https://oeis.org/A112028/internal is the same sequence with the initial zero omitted. Its comment by Peter Bala dated March 29, 2023 proposes the p^5 congruence and attributes the weaker p^3 result to Coster, Theorem 4. Thus the conjecture predates the 2024 asymptotic comment in A375178; the bibkey year does not assert priority.

Mathematical sources and scope

Meštrović, sections 4 and 10, surveys related harmonic and binomial congruences without stating this particular p^5 sum. Zhao, “Wolstenholme type theorem for multiple harmonic sums”, https://arxiv.org/abs/math/0301252, Lemma 2.2, Theorem 3.2, equation (22), and Lemma 3.3 cover the needed cubic harmonic vanishing, double harmonic vanishing, shuffle, and reversal. These are published prerequisites used in the repository derivation.

Coster’s Theorem 4 (https://doi.org/10.1007/BFb0091139) is cited through OEIS’s attribution; its full text and precise scope remain ASSUMED-UNVERIFIED.

The module proves the universal p^5 statement using local harmonic lemmas and mathlib’s finite-field power sums. It makes no assertion of global priority and does not address either of the two further OEIS conjectures.