{"paper":{"title":"On binomial coefficients modulo squares of primes","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Darij Grinberg","submitted_at":"2017-12-06T09:14:40Z","abstract_excerpt":"We give elementary proofs for the Apagodu-Zeilberger-Stanton-Amdeberhan-Tauraso congruences $$\\sum\\limits_{n=0}^{p-1}\\dbinom{2n}{n} \\equiv\\eta_{p}\\mod p^{2},$$ $$\\sum\\limits_{n=0}^{rp-1}\\dbinom{2n}{n} \\equiv\\eta_{p}\\sum\\limits_{n=0}^{r-1}\\dbinom {2n}{n}\\mod p^{2}$$ and $$\\sum\\limits_{n=0}^{rp-1}\\sum\\limits_{m=0}^{sp-1}\\dbinom{n+m}{m}^{2} \\equiv\\eta_{p} \\sum\\limits_{m=0}^{r-1}\\sum\\limits_{n=0}^{s-1}\\dbinom{n+m}{m}^2\\mod p^2,$$ where $p$ is an odd prime, $r$ and $s$ are nonnegative integers, and $\\eta_{p}= \\begin{cases} 0, &\\text{if }p\\equiv0\\mod 3;\\\\ 1, & \\text{if }p\\equiv1\\mod 3;\\\\ -1, &\\text{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.02095","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}