OpenAlex ·
A two-step supercongruence for an Apéry-like sequence
- Published
- Source
- OpenAlex
- Published in
- arXiv (Cornell University)
- DOI
- 10.48550/arxiv.2609.23355
- Original
- Read it at the publisher
- Topics
Abstract
Let $G_n=\sum_{k=0}^n4^k\binom{2n-2k}{n-k}^{2}\binom{2k}{k}$. Zhi-Hong Sun conjectured that, for primes $p\equiv3\pmod4$, positive odd integers $m$, and $r\ge2$, the two-step congruence $G_{(mp^r-1)/2}\equiv p^2G_{(mp^{r-2}-1)/2}\pmod {p^{2r-1}}$ holds. We prove the stronger valuation statement $G_{(p^2M-1)/2}-p^2G_{(M-1)/2}\in p^{2v_p(M)+3}\mathbb Z_p$ 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.
The authors’ abstract, as arXiv lists it. arXiv’s listings, abstracts included, are free for anyone to reuse.
Read it on ReadSearch
Each opens after you create an account, and brings you back to this paper.
More in Combinatorics
Get the few papers worth your time in Combinatorics every morning.
ReadSearch reads the new papers in your field each day and brings you the few worth reading, each with its key finding in a sentence.