{"paper":{"title":"Conjectures and results on $x^2$ mod $p^2$ with $4p=x^2+dy^2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2011-03-22T17:46:57Z","abstract_excerpt":"Given a squarefree positive integer $d$, we want to find integers (or rational numbers with denominators not divisible by large primes) $a_0,a_1,a_2,\\ldots$ such that for sufficiently large primes $p$ we have $\\sum_{k=0}^{p-1}a_k\\equiv x^2-2p$ (mod $p^2$) if $4p=x^2+dy^2$ (and $4\\nmid x$ if $d=1$), and $\\sum_{k=0}^{p-1}a_k\\equiv 0$ (mod $p^2$) if $(\\frac{-d}p)=-1$. In this paper we give a survey of conjectures and results on this topic and point out the connection between this problem and series for $1/\\pi$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.4325","kind":"arxiv","version":11},"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"}