{"paper":{"title":"Perfect powers in sequences of polygonal numbers","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andrzej D\\k{a}browski, G\\\"okhan Soydan, Paul M. Voutier, Salah Eddine Rihane","submitted_at":"2026-06-26T16:17:48Z","abstract_excerpt":"Let $P_s(n)$ denote the $n$-th $s$-gonal number. Consider the Diophantine equation $P_{s}(n) = t^{m}$ for integers $n, s, t$ and $m > 2$. All solutions to this equation are known for $m>2$ and $s\\in\\{3,5,6,8,10,20\\}$. Here we extend these results to the cases $s = 2k+4$ (where $k = 4,6$ or $5 \\leq k \\leq 97$ is a prime number) and $s = k+4$ (where $k = 9,15$ or $3 \\leq k \\leq 97$ is a prime number).\n  The proofs of our results use the modular and hypergeometric methods, linear forms in logarithms and extensive calculations. We were unable to completely solve the above Diophantine equations, bu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.28227","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.28227/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}