A two-step supercongruence for an Apéry-like sequence
Abstract
Let Gn=∑k=0n4kbinom2n−2kn−k2binom2kk. Zhi-Hong Sun conjectured that, for primes p≡3pmod4, positive odd integers m, and r≥2, the two-step congruence G(mpr−1)/2≡ p2G(mpr−2−1)/2pmod p2r−1 holds. We prove the stronger valuation statement G(p2M−1)/2−p2G(M−1)/2∈ p2vp(M)+3ℤp for every positive odd M. The proof converts the sum to a terminating 3F2, constructs a cancelled digit-transfer operator, and identifies a two-dimensional analytic quotient of its cubic difference operator. The quotient operator has characteristic polynomial X2−p2; its second trace is evaluated through the Gross--Koblitz formula and Greene's finite-field Dixon identity. A logarithmic-loss Green inverse converts this spectral identity into an actual integral analytic primitive. The exceptional prime $3$ requires a finite exact PARI/GP certificate, while the infinite tail is bounded symbolically. Thus the computation is finite, reproducible, and separated from the uniform part of the proof.
Related
- Galois groups of Apéry-like series modulo primes
- Identities and congruences involving orthogonal polynomials and Apéry-like numbers
- An Infinite Family of Apéry-like Recurrences for Euler's Constant
- Weighted bilinear identities and supercongruences for Apéry-like polynomials
- Sharp Asymptotic Minimaxity of the Gavrilov-Benjamini-Sarkar Step-Down Testing Procedure in Sparse Gaussian Sequence Models
BibTeX
@misc{zheng2026twostep,
title = {{A two-step supercongruence for an Apéry-like sequence}},
author = {Huimin Zheng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23355},
eprint = {2609.23355},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23355},
}
RIS
TY - UNPB
TI - A two-step supercongruence for an Apéry-like sequence
AU - Huimin Zheng
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23355
UR - https://arxiv.org/abs/2609.23355
AB - Let G_n=∑_k=0^n4^kbinom2n−2kn−k^2binom2kk. Zhi-Hong Sun conjectured that, for primes p≡3pmod4, positive odd integers m, and r≥2, the two-step congruence G_(mp^r−1)/2≡ p^2G_(mp^r−2−1)/2pmod p^2r−1 holds. We prove the stronger valuation statement G_(p^2M−1)/2−p^2G_(M−1)/2∈ p^2v_p(M)+3ℤ_p for every positive odd M. The proof converts the sum to a terminating _3F_2, constructs a cancelled digit-transfer operator, and identifies a two-dimensional analytic quotient of its cubic difference operator. The quotient operator has characteristic polynomial X^2−p^2; its second trace is evaluated through the Gross--Koblitz formula and Greene's finite-field Dixon identity. A logarithmic-loss Green inverse converts this spectral identity into an actual integral analytic primitive. The exceptional prime $3$ requires a finite exact PARI/GP certificate, while the infinite tail is bounded symbolically. Thus the computation is finite, reproducible, and separated from the uniform part of the proof.
ER -