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Zhi-Hong Sun's Conjecture 3.6 on two Apéry-like binomial sums

Let fn=∑j=0n(nj)3f_n=\sum_{j=0}^n\binom{n}{j}^3 and Qn=∑k=0n(nk)(−8)n−kfkQ_n=\sum_{k=0}^n\binom{n}{k}(-8)^{n-k}f_k. Is it true that, for every odd prime pp,∑n=0p−1(2nn)Qn(−32)n≡∑n=0p−1(2nn)Qn64n≡{4x2−2p,p=x2+2y2≡1,3(mod8),0,p≡5,7(mod8)(modp2)?\sum_{n=0}^{p-1}\frac{\binom{2n}{n}Q_n}{(-32)^n}\equiv\sum_{n=0}^{p-1}\frac{\binom{2n}{n}Q_n}{64^n}\equiv\begin{cases}4x^2-2p,&p=x^2+2y^2\equiv1,3\pmod8,\\0,&p\equiv5,7\pmod8\end{cases}\pmod{p^2}?Here x,yx,y are integers in the first case. This is Conjecture 3.6 of Zhi-Hong Sun, AIMS Mathematics 7(2) (2022), p. 2742, published online on 18 November 2021. The question includes both sums and the primes 3 and 5.

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-assisted
Method
Argument
Field
Supercongruences, Apéry-like sequences and modular forms
Posed by
Zhi-Hong Sun, Conjecture 3.6, AIMS Mathematics 7(2) (2022), p. 2742; DOI 10.3934/math.2022153. Published online 18 November 2021.
Year posed
2021
Years open
5y
Solved
2026-10
Model
OpenAI Codex; GPT-6.1 Sol
Vendor
OpenAI
Collaborators
Oleksiy Babanskyy
Verification
Unreviewed
Publication
Announced
Significance
5 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1 claims the complete stated conjecture: both sums, every odd prime and modulus p squared. For inert primes, a finite differential invariant lifts Sun's mod-p result, with an autonomous singular treatment at p=5. For split primes, certified eta identities, a generic integral Hecke polynomial, an explicit degree bound and three CM matrix cases yield the result. An independently sourced coefficient congruence removes the degree-p boundary; p=3 is checked directly. The paper credits Sun's mod-p input, the Straub-Gorodetsky endpoint and Beukers's modular framework. It does not claim a uniform mod-p-cubed extension: at p=7 the two sums are 294 and 49 modulo 343. The full conjecture is claimed, but authoritative mathematical review is pending.

What the AI did

AI assisted mathematical exploration, proof criticism, source comparison, manuscript drafting, exact computation and repository preparation under the author's direction. OpenAI Codex and GPT-6.1 Sol assisted consolidation and separate checks of the proof and companion. External-agent material was also examined; its complete model provenance is not asserted. Individual proof steps are not attributed to particular systems. The systems had access to related arguments, so their checks are not independent human peer review or formal verification. This assistance is disclosed in the manuscript and AI_USE.md. The author retains responsibility for the claims and publication.

Verification

Checked by this site on 4 October 2026. The statement was compared with Conjecture 3.6 as printed on p. 2742 of Sun's AIMS Mathematics paper and matches it in full (both sums, every odd prime, modulus p squared). A separate script written here from the binomial definitions found no counterexample at any prime below 260. The author's companion recomputes both sides for the primes to 1999 and certifies the finite steps of the argument; these are finite checks, and the all-primes theorem rests on the written modular-forms proof, which was not refereed here. The author-side reviews are by AI models. Prior work: citing literature (OpenAlex, Semantic Scholar), Z.-H. Sun's arXiv papers and Sun-Ye's 2024 preprint were searched and no earlier proof was found; the paywalled final texts of Mao (2026), Sun-Ye (2025) and Sun (2026) were not read.

Sources

Submitted by Oleksiy Babanskyy on

Changelog2 changes

Discussion