{"paper":{"title":"Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Romeo Mestrovic","submitted_at":"2011-11-13T19:32:11Z","abstract_excerpt":"In 1862 Wolstenholme proved that for any prime $p\\ge 5$ the numerator of the fraction $$ 1+\\frac 12 +\\frac 13+...+\\frac{1}{p-1}\n  $$ written in reduced form is divisible by $p^2$, $(2)$ and the numerator of the fraction\n  $$ 1+\\frac{1}{2^2} +\\frac{1}{3^2}+...+\\frac{1}{(p-1)^2}\n  $$ written in reduced form is divisible by $p$. The first of the above congruences, the so called {\\it Wolstenholme's theorem}, is a fundamental congruence in combinatorial number theory. In this article, consisting of 11 sections, we provide a historical survey of Wolstenholme's type congruences and related problems. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.3057","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"}