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Sun's Conjecture 3.5 on an Apéry-like supercongruence

Let an=∑j=0n(nj)2(2jj)a_n=\sum_{j=0}^{n}\binom{n}{j}^{2}\binom{2j}{j}. For every prime p>3p>3, is it true that
(−1)(p−1)/2∑n=0p−1(2nn)an54n≡{4x2−2p,p=x2+3y2≡1(mod3),0,p≡2(mod3)(modp2)? (-1)^{(p-1)/2}\sum_{n=0}^{p-1}\binom{2n}{n}\frac{a_n}{54^n}\equiv \begin{cases} 4x^2-2p,&p=x^2+3y^2\equiv1\pmod{3},\\ 0,&p\equiv2\pmod{3} \end{cases} \pmod{p^2}?
This is Conjecture 3.5 on p. 2742 of Zhi-Hong Sun's AIMS Mathematics paper, DOI 10.3934/math.2022153. It is distinct from Theorem 3.5 in the same article, which gives a congruence modulo pp.

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 3.5, AIMS Mathematics 7(2) (2022), p. 2742; published online 18 November 2021. DOI: 10.3934/math.2022153.
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 modulo p2p^2 for every prime p>3p>3, covering both residue classes modulo 3, with p=5p=5 treated directly. For inert primes, a finite quadratic differential invariant lifts Sun's published modulo-pp vanishing. For split primes, certified modular periods, an integral Hecke polynomial with a degree bound and complex multiplication give the evaluation.

The Franel sum with denominator 50 is evaluated separately for p>5p>5 and credited to Sun's different 2018 Conjecture 3.5. No modulo-p3p^3 result, resolution of adjacent conjectures, or first-proof claim is made. Candidate reflects pending authoritative review of a claimed complete proof of this specific question.

What the AI did

Under the author's direction, AI systems contributed substantially to mathematical exploration, proof criticism, source comparison, manuscript drafting, exact computation and repository preparation. AI_USE.md names OpenAI Codex and GPT-6.1 Sol for consolidation and separate checks. Material supplied through an external agent was also examined; its complete model provenance is not asserted. Individual proof steps are not attributed to particular models. The author is responsible for the mathematical claims and publication.

Verification

Checked by this site on 4 October 2026. The statement was compared with Conjecture 3.5 as printed on p. 2742 of Sun's AIMS Mathematics paper and matches it in full (every prime p > 3, both residue classes mod 3, 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