{"paper":{"title":"On geometric progressions on hyperelliptic curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Mohamed Alaa, Mohammad Sadek","submitted_at":"2016-02-18T16:03:37Z","abstract_excerpt":"Let $C$ be a hyperelliptic curve over $\\mathbb Q$ described by $y^2=a_0x^n+a_1x^{n-1}+\\ldots+a_n$, $a_i\\in\\mathbb Q$. The points $P_{i}=(x_{i},y_{i})\\in C(\\mathbb{Q})$, $i=1,2,...,k,$ are said to be in a geometric progression of length $k$ if the rational numbers $x_{i}$, $i=1,2,...,k,$ form a geometric progression sequence in $\\mathbb Q$, i.e., $x_i=pt^{i}$ for some $p,t\\in\\mathbb Q$. In this paper we prove the existence of an infinite family of hyperelliptic curves on which there is a sequence of rational points in a geometric progression of length at least eight."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.05850","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"}