{"paper":{"title":"Partition regularity of generalised Fermat equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Sofia Lindqvist","submitted_at":"2016-06-23T14:44:43Z","abstract_excerpt":"Let $\\alpha,\\beta,\\gamma\\in\\mathbb{N}$. We prove that given an $r$-colouring of $\\mathbb{F}_p$ with $p$ prime, there are more than $c_{r,\\alpha,\\beta,\\gamma} p^2$ solutions to the equation $x^\\alpha+y^\\beta=z^\\gamma$ with all of $x,y,z$ of the same colour. Here $c_{r,\\alpha,\\beta,\\gamma}>0$ is some constant depending on the number of colours and the exponents in the equation. This is already a new result for $\\alpha=\\beta=1$ and $\\gamma=2$, that is to say for the equation $x+y=z^2$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.07334","kind":"arxiv","version":3},"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"}