{"paper":{"title":"Arithmetic properties of polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Yong Zhang, Zhongyan Shen","submitted_at":"2017-06-12T01:38:36Z","abstract_excerpt":"In this paper, first, we prove that the Diophantine system \\[f(z)=f(x)+f(y)=f(u)-f(v)=f(p)f(q)\\] has infinitely many integer solutions for $f(X)=X(X+a)$ with nonzero integers $a\\equiv 0,1,4\\pmod{5}$. Second, we show that the above Diophantine system has an integer parametric solution for $f(X)=X(X+a)$ with nonzero integers $a$, if there are integers $m,n,k$ such that \\[\\begin{cases} \\begin{split} (n^2-m^2) (4mnk(k+a+1) + a(m^2+2mn-n^2)) &\\equiv0\\pmod{(m^2+n^2)^2},\\\\ (m^2+2mn-n^2) ((m^2-2mn-n^2)k(k+a+1) - 2amn) &\\equiv0 \\pmod{(m^2+n^2)^2}, \\end{split} \\end{cases}\\] where $k\\equiv0\\pmod{4}$ when"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.03433","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"}