{"paper":{"title":"On the number of distinct prime factors of a sum of super-powers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Paolo Leonetti, Salvatore Tringali","submitted_at":"2015-11-27T20:20:31Z","abstract_excerpt":"Given $k, \\ell \\in {\\bf N}^+$, let $x_{i,j}$ be, for $1 \\le i \\le k$ and $0 \\le j \\le \\ell$, some fixed integers, and define, for every $n \\in {\\bf N}^+$, $s_n := \\sum_{i=1}^k \\prod_{j=0}^\\ell x_{i,j}^{n^j}$. We prove that the following are equivalent:\n  (a) There are a real $\\theta > 1$ and infinitely many indices $n$ for which the number of distinct prime factors of $s_n$ is greater than the super-logarithm of $n$ to base $\\theta$.\n  (b) There do not exist non-zero integers $a_0,b_0,\\ldots,a_\\ell,b_\\ell $ such that $s_{2n}=\\prod_{i=0}^\\ell a_i^{(2n)^i}$ and $s_{2n-1}=\\prod_{i=0}^\\ell b_i^{(2"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.08784","kind":"arxiv","version":3},"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"}