{"id":"hexagon/2609.00130","pdf":"https://hexagonmath.org/pdf/2609.00130v1","url":"https://hexagonmath.org/abs/2609.00130v1","tags":["msc:11D45"],"score":0,"title":"Primitive Integer Solutions of x^4 + 4y^3 + z^3 = 0","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Avraham Eisenberg"],"reviews":0,"subject":"math.NT","summary":"We determine all primitive integer solutions of the generalized Fermat equation x^4 + 4y^3 + z^3 = 0. Here primitive means gcd(x,y,z)=1. We prove that the only primitive solutions are (x,y,z) = (1,0,-1), (-1,0,-1). The proof begins with a fourth-power descent in the pure cubic field Q(∛2), reducing the problem to the rational points on two explicit smooth plane quartics of genus 3. These quartics admit maps to elliptic curves over Q(∛2). Exact 2-descent determines the relevant Mordell-Weil ranks and finite-index subgroups. A complete reduction sieve at 127 leaves four residue disks, and an…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["ChatGPT"],"discussed":0,"published":"2026-09-29","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"none"}