{"paper":{"title":"A congruence involving alternating harmonic sums modulo $p^{\\alpha}q^{\\beta}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Tianxin Cai, Zhongyan Shen","submitted_at":"2015-03-11T02:43:32Z","abstract_excerpt":"In 2014, Wang and Cai established the following harmonic congruence for any odd prime $p$ and positive integer $r$, \\begin{equation*}\n  \\sum\\limits_{i+j+k=p^{r}\\atop{i,j,k\\in \\mathcal{P}_{p}}}\\frac{1}{ijk}\\equiv-2p^{r-1}B_{p-3} ~(\\bmod ~ p^{r}), \\end{equation*} where $\\mathcal{P}_{n}$ denote the set of positive integers which are prime to $n$. In this note, we obtain the congruences for distinct odd primes $p,~q$ and positive integers $\\alpha,~\\beta$, \\begin{equation*}\n  \\sum\\limits_{i+j+k=p^{\\alpha}q^{\\beta}\\atop{i,j,k\\in\\mathcal{P}_{pq}\\atop{i\\equiv j\\equiv k\\equiv 1\\pmod{2}}}}\\frac{1}{ijk}\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.03154","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"}