{"paper":{"title":"Quartic, octic residues and binary quadratic forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Hong Sun","submitted_at":"2011-08-15T16:14:02Z","abstract_excerpt":"Let $\\Bbb Z$ be the set of integers, and let $(m,n)$ be the greatest common divisor of integers $m$ and $n$. Let $p\\equiv 1\\mod 4$ be a prime, $q\\in\\Bbb Z$, $2\\nmid q$ and $p=c^2+d^2=x^2+qy^2$ with $c,d,x,y\\in\\Bbb Z$ and $c\\e 1\\mod 4$. Suppose that $(c,x+d)=1$ or $(d,x+c)$ is a power of 2. In the paper, by using the quartic reciprocity law we determine $q^{[p/8]}\\mod p$ in terms of $c,d,x$ and $y$, where $[\\cdot]$ is the greatest integer function. We also determine $\\big(\\frac{b+\\sqrt{b^2+4^{\\alpha}}}2\\big)^{\\frac{p-1}4}\\mod p$ for odd $b$ and $(2a+\\sqrt{4a^2+1})^{\\f{p-1}4}\\mod p$ for $a\\in\\Bb"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.3027","kind":"arxiv","version":4},"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"}