{"paper":{"title":"Some congruences related to the q-Fermat quotients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Victor J. W. Guo","submitted_at":"2013-12-23T12:43:00Z","abstract_excerpt":"We give q-analogues of the following congruences by Z.-W. Sun: \\sum_{k=1}^{p-1}\\frac{D_k}{k} \\equiv -\\frac{2^{p-1}-1}{p} \\pmod p,\\\\ \\sum_{k=1}^{p-1}\\frac{H_k}{k 2^k}\\equiv 0 \\pmod{p},\\quad p\\geqslant 5, where p is a prime, D_n=\\sum_{k=0}^{n}{n+k\\choose 2k}{2k\\choose k} are the Delannoy numbers, and H_n=\\sum_{k=1}^n\\frac{1}{k} are the harmonic numbers. We also prove that, for any positive integer m and prime p>m+1, \\sum_{1\\leqslant k_1\\leqslant \\cdots \\leqslant k_m\\leqslant p-1}\\frac{1}{k_1\\cdots k_m 2^{k_m}} \\equiv\\frac{1}{2}\\sum_{k=1}^{p-1}\\frac{(-1)^{k-1}}{k^m} \\pmod p, which is a multiple g"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.6541","kind":"arxiv","version":1},"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"}