{"paper":{"title":"Multivariate Ap\\'ery numbers and supercongruences of rational functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Armin Straub","submitted_at":"2014-01-05T00:21:36Z","abstract_excerpt":"One of the many remarkable properties of the Ap\\'ery numbers $A (n)$, introduced in Ap\\'ery's proof of the irrationality of $\\zeta (3)$, is that they satisfy the two-term supercongruences \\begin{equation*}\n  A (p^r m) \\equiv A (p^{r - 1} m) \\pmod{p^{3 r}} \\end{equation*} for primes $p \\geq 5$. Similar congruences are conjectured to hold for all Ap\\'ery-like sequences. We provide a fresh perspective on the supercongruences satisfied by the Ap\\'ery numbers by showing that they extend to all Taylor coefficients $A (n_1, n_2, n_3, n_4)$ of the rational function \\begin{equation*}\n  \\frac{1}{(1 - x_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.0854","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"}