{"paper":{"title":"Standard morphisms and Pythagorean triples","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Andr\\'e Carvalho, Jo\\~ao Ara\\'ujo","submitted_at":"2026-08-12T12:06:46Z","abstract_excerpt":"Let $m\\geq 1$, let $f:\\mathbb N\\to\\mathbb Z/m\\mathbb Z$ be a standard morphism and let $T(m)$ be the least integer $N$ such that every such $f$ admits a primitive monochromatic Pythagorean triple with hypotenuse at most $N$. The aim of this note is to prove that every standard morphism has infinitely many identity-valued Pythagorean triples and infinitely many primitive monochromatic Pythagorean triples. Thus the qualitative part of Problem~4.3 of Eliahou, Fromentin, Marion-Poty and Robilliard is solved for every $m$. Moreover, $T(m)$ is finite, the morphism $n\\mapsto v_3(n)\\pmod m$ has least "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.11975","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/2608.11975/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"}