{"paper":{"title":"A Pair of Diophantine Equations and Fibonacci-Like Sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Hung Viet Chu, Nathanael Johnson, Rishabh Gulecha, Sicheng Guo, Steven J. Miller, Yeju Shin","submitted_at":"2025-09-01T21:27:08Z","abstract_excerpt":"Given two relatively prime numbers $a$ and $b$, it is known that exactly one of the two Diophantine equations has a nonnegative integral solution $(x,y)$: $$\n  ax + by \\ =\\ \\frac{(a-1)(b-1)}{2}\\quad \\mbox{ and }\\quad 1 + ax + by \\ =\\ \\frac{(a-1)(b-1)}{2}. $$ Furthermore, the solution is unique. This paper surveys recent results on finding the solution and determining which equation is used when $a$ and $b$ are taken from certain sequences. We contribute to the literature by finding $(x,y)$ when $a$ and $b$ are consecutive terms of sequences having the Fibonacci recurrence and arbitrary initial"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.01781","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2509.01781/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"}