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Zhi-Hong Sun's Conjecture 3.7: two Apéry-like sums at discriminant -20

Let fn=∑j=0n(nj)3f_n=\sum_{j=0}^n\binom{n}{j}^3, an=∑j=0n(nj)2(2jj)a_n=\sum_{j=0}^n\binom{n}{j}^2\binom{2j}{j}, Qn=∑j=0n(nj)(−8)n−jfjQ_n=\sum_{j=0}^n\binom{n}{j}(-8)^{n-j}f_j and εp=(−1)(p−1)/2\varepsilon_p=(-1)^{(p-1)/2}. For every prime p≠2,5p\ne2,5, is it true thatεp∑n=0p−1(2nn)an20n≡εp∑n=0p−1(2nn)Qn(−16)n≡{4x2−2p,p=x2+5y2, p≡1,9(mod20),2x2−2p,2p=x2+5y2, p≡3,7(mod20),0,p≡11,13,17,19(mod20)(modp2)?\varepsilon_p\sum_{n=0}^{p-1}\frac{\binom{2n}{n}a_n}{20^n}\equiv\varepsilon_p\sum_{n=0}^{p-1}\frac{\binom{2n}{n}Q_n}{(-16)^n}\equiv\begin{cases}4x^2-2p,&p=x^2+5y^2,\ p\equiv1,9\pmod{20},\\2x^2-2p,&2p=x^2+5y^2,\ p\equiv3,7\pmod{20},\\0,&p\equiv11,13,17,19\pmod{20}\end{cases}\pmod{p^2}?Here x,yx,y are integers in the representation cases. This is Conjecture 3.7 of Zhi-Hong Sun, AIMS Mathematics 7(2) (2022), p. 2742, published online on 18 November 2021.

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-assisted
Method
Argument
Field
Supercongruences, modular forms and elliptic curves
Posed by
Zhi-Hong Sun, Conjecture 3.7, 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 conjecture: both signed sums, every prime except 2 and 5, all representation classes, and modulus p squared. Split primes use two modular periods, an integral Hecke polynomial and separate finite specializations, including the degree-p endpoint and nonprincipal CM sign. For inert primes, a degree-five endomorphism with square [-5] forces Hasse vanishing, then a differential invariant lifts each original sum. The prime 3 is checked directly. The paper credits Sun's mod-p and Legendre antecedents, Straub's coefficient congruence and Beukers's framework. The stronger Sun-Ye quartic congruence concerns an auxiliary sum; the available finite transports alone have only mod-p precision. No uniform mod-p-cubed equality is claimed: at p=7 the unsigned target sums are 192 and 290 modulo 343.

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 3.7 as printed on p. 2742 of Sun's AIMS Mathematics paper and matches it in full (both signed sums, every prime other than 2 and 5, all representation classes, 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