{"paper":{"title":"When is $a^{n} + 1$ the sum of two squares?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Aaron Schmitt, Emily Stamm, Greg Dresden, Jeremy Rouse, Kylie Hess, Pan Yue, Saimon Islam, Terrin Warren","submitted_at":"2016-09-14T19:50:18Z","abstract_excerpt":"Using Fermat's two squares theorem and properties of cyclotomic polynomials, we prove assertions about when numbers of the form $a^{n}+1$ can be expressed as the sum of two integer squares. We prove that $a^n + 1$ is the sum of two squares for all $n \\in \\mathbb{N}$ if and only if $a$ is a perfect square. We also prove that for $a\\equiv 0,1,2\\pmod{4},$ if $a^{n} + 1$ is the sum of two squares, then $a^{\\delta} + 1$ is the sum of two squares for all $\\delta | n, \\ \\delta>1$. Using Aurifeuillian factorization, we show that if $a$ is a prime and $a\\equiv 1 \\pmod{4}$, then there are either zero or"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.04391","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"}