{"paper":{"title":"Proof of a recent conjecture of Z.-W. Sun","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Song Guo, Victor J. W. Guo","submitted_at":"2016-04-18T07:22:38Z","abstract_excerpt":"The polynomials $d_n(x)$ are defined by \\begin{align*} d_n(x) &= \\sum_{k=0}^n{n\\choose k}{x\\choose k}2^k. \\end{align*} We prove that, for any prime $p$, the following congruences hold modulo $p$: \\begin{align*} \\sum_{k=0}^{p-1}\\frac{{2k\\choose k}}{4^k} d_k\\left(-\\frac{1}{4}\\right)^2 &\\equiv \\begin{cases} 2(-1)^{\\frac{p-1}{4}}x,&\\text{if $p=x^2+y^2$ with $x\\equiv 1\\pmod{4}$,} 0,&\\text{if $p\\equiv 3\\pmod{4}$,} \\end{cases} [5pt] \\sum_{k=0}^{p-1}\\frac{{2k\\choose k}}{4^k} d_k\\left(-\\frac{1}{6}\\right)^2 &\\equiv 0, \\quad\\text{if $p>3$,} [5pt] \\sum_{k=0}^{p-1}\\frac{{2k\\choose k}}{4^k} d_k\\left(\\frac{1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.05019","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"}