{"paper":{"title":"A polynomial variant of a problem of Diophantus and its consequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alan Filipin, Ana Jurasi\\'c","submitted_at":"2017-05-25T14:21:45Z","abstract_excerpt":"We prove that every Diophantine quadruple in $\\mathbb{R}[X]$ is regular. More precisely, we prove that if $\\{a, b, c, d\\}$ is a set of four non-zero polynomials from $\\mathbb{R}[X]$, not all constant, such that the product of any two of its distinct elements increased by $1$ is a square of a polynomial from $\\mathbb{R}[X]$, then $$(a+b-c-d)^2=4(ab+1)(cd+1).$$\n  One consequence of this result is that there does not exist a set of four non-zero polynomials from $\\mathbb{Z}[X]$, not all constant, such that a product of any two of them increased by a positive integer $n$, which is not a perfect sq"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.09194","kind":"arxiv","version":2},"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"}