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Sun's supercongruence for an Apéry-like sequence

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theory

Contributed by Oleksiy Babanskyy ↗

Version 1 / Oct 27, 2026 / CC BY 4.0

Abstract

Let an=∑j=0n(nj)2(2jj)a_n=\sum_{j=0}^n\binom{n}{j}^2\binom{2j}{j}. We prove that, for every prime p>3p>3, the sum (−1)(p−1)/2∑n=0p−1(2nn)an/54n(-1)^{(p-1)/2}\sum_{n=0}^{p-1}\binom{2n}{n}a_n/54^n is congruent modulo p2p^2 to 4x2−2p4x^2-2p when p=x2+3y2≡1(mod3)p=x^2+3y^2\equiv1\pmod3, and to zero when p≡2(mod3)p\equiv2\pmod3. For the latter primes a quadratic differential invariant of the truncated generating polynomial lifts Sun's congruence modulo pp. For the former primes we certify two modular periods directly by identities of weight eight, construct a Hecke polynomial with an explicit degree bound, and specialize a finite polynomial identity at a complex multiplication point. The coefficient at degree pp and both parity cases of the complex multiplication matrices are treated explicitly. A corresponding evaluation of the Franel sum with denominator 5050 follows by the same arguments.

Provenance statement

This work used substantial AI assistance under the author's direction for mathematical exploration, proof criticism, source comparison, manuscript drafting, exact computation and repository preparation. OpenAI Codex and GPT-6.1 Sol assisted with consolidation and separate checks of the proof and companion. Material supplied by an external AI agent was also examined; complete model provenance for that material is not asserted. Individual proof steps are not attributed to particular AI systems. The manuscript incorporates comments from Zhi-Hong Sun, acknowledged in the paper. The exact-arithmetic companion provides reproducible computational checks. These checks do not constitute independent human peer review or end-to-end formal verification. Oleksiy Babanskyy is responsible for the manuscript and its claims. This deposit reproduces the revised manuscript available as aiXiv 2610.00155 v3: https://aixiv.online/abs/2610.00155v3 Source, companion code and reproduction instructions: https://github.com/aconsciousfractal/Sun-3-5-Supercongruence/tree/7c3f3a4a956dac3c28c6c6026c650419eefe6d90

Tools used

OpenAI
CodexVersion sol 6.1
Python Software Foundatio
Python

References

  1. Zhi-Hong Sun, Supercongruences involving Apéry-like numbers and binomial coefficients , AIMS Mathematics 7 (2) (2022), 2729--2781. doi:10.3934/math.2022153 (https://doi.org/10.3934/math.2022153).DOI
  2. Zhi-Hong Sun, Congruences for sums involving Franel numbers , International Journal of Number Theory 14 (1) (2018), 123--142. Author's full text (https://maths.hytc.edu.cn/__local/3/45/77/77F7497DDFD531E8688ED13C0D0_262306C8_24EB4.pdf).link
  3. Frits Beukers, Supercongruences using modular forms , arXiv:2403.03301v3 (https://arxiv.org/abs/2403.03301v3) (2025).arXiv
  4. Jeremy Rouse and John J. Webb, On spaces of modular forms spanned by eta-quotients , arXiv:1311.1460v3 (https://arxiv.org/abs/1311.1460v3) (2014).arXiv
  5. NIST Digital Library of Mathematical Functions, Section 23.18, Modular Transformations , equations 23.18.5--23.18.7. https://dlmf.nist.gov/23.18.link

Version history

  1. v1Submitted by Oleksiy BabanskyyInitial depositCurrentOct 27, 2026