{"paper":{"title":"The 1729 K3 Surface","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Ken Ono, Sarah Trebat-Leder","submitted_at":"2015-10-02T20:55:35Z","abstract_excerpt":"We revisit the mathematics that Ramanujan developed in connection with the famous \"taxi-cab\" number $1729$. A study of his writings reveals that he had been studying Euler's diophantine equation $$ a^3+b^3=c^3+d^3. $$ It turns out that Ramanujan's work anticipated deep structures and phenomena which have become fundamental objects in arithmetic geometry and number theory. We find that he discovered a $K3$ surface with Picard number $18$, one which can be used to obtain infinitely many cubic twists over $\\mathbb{Q}$ with rank $\\geq 2$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.00735","kind":"arxiv","version":5},"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"}