bibkey: oeis2026a397436
authors: Peter Bala; OEIS Foundation Inc.
year: 2026
title: OEIS A397436 revision 44, Conjecture 3 on integer-parameter binomial products
doi: null
url: https://oeis.org/A397436/internal
claim: The unrestricted same-r supercongruence for inclusive truncated products of signed generalized binomial coefficients.
strata_touched:
  - D5/S3/ArithSums/A397436GeneralSupercongruenceRefutation
license: citation-only
triage: anchor

A397436 Conjecture 3

Primary statement

OEIS A397436 revision 44 states:

3. More generally, let m_1, …, m_r be integers and M a positive integer. Define u(n) = Sum_{k = 0..Mn} binomial(m_1n, k)… binomial(m_rn, k). The supercongruence u(p^r) == u(p^(r-1)) (mod p^(3r)) holds for all primes p >= 5 and all positive integers r (the present sequence is the case M = 1, m_1 = -1 and m_2 = -3).

The entry credits Peter Bala; item 3 has no separate signature. The short quotation is attributed to OEIS under https://oeis.org/LICENSE. No external proof code is copied.

Verified locator

The primary URL is https://oeis.org/A397436/internal. The qualified complete revision-44 internal export has SHA-256 131681bfb4e331f9f4ec7fecbe2452a8d4958508d72365bfde54b688604d1413; its JSON export at https://oeis.org/A397436?fmt=json has SHA-256 86a946ee702c326e32be268f376379bc3c0e98dbf0f3cf95fb2e3d192679b6ed. The public source and exact statement registration are at https://github.com/the-omega-institute/trureturing/issues/23456 and https://github.com/the-omega-institute/trureturing/issues/23456#issuecomment-6485402841.

Domain and convention

The literal same-r reading uses r for both the number of factors and the lift exponent. The sum includes k=0 and k=M*n. Each m_i is an arbitrary integer; the full entry supplies no negative-only condition or two-factor minimum. Restrictions belonging to Conjectures 1 and 2 are not imported. The parenthetical two-factor example leaves author intent about separate factor and lift indices unresolved. The witness sets both indices to one, so also refutes an unrestricted reading with independent positive indices. No claim about an unprinted restriction is made.

The direct cross-reference A333592 explicitly uses C(-n,k)=(-1)^k*C(n+k-1,k). Pinned Mathlib’s Ring.choose on Int has this negative-upper convention, satisfies the descending-factorial identity, and equals Nat.choose for natural upper arguments by Ring.choose_natCast. The witness uses only positive upper arguments while the quantified statement retains all signed parameters.

Literature scope

The independent qualification read the full A397436, A005810 and A333592 entries, repository declarations including private helpers, the published open-problems page, bounded all-state GitHub exact-ID and supercongruence searches, and Blanco Gomez’s original English text at https://doi.org/10.5281/zenodo.23426707. That text’s balanced factorial/Gamma ratio results do not supply the full arbitrary truncated-product assertion. No complete settlement or competing owner was identified in that scope. This is bounded absence evidence, with no worldwide openness, priority, novelty or complete alias-search claim. The proof here is repo-derived.