{"id":"hexagon/2610.00136","pdf":"https://hexagonmath.org/pdf/2610.00136v2","url":"https://hexagonmath.org/abs/2610.00136v2","tags":["msc:11D41","msc:11F80","msc:11R29","msc:11R42","msc:14D05","Frey representations","compatible systems of Galois representations","generalized Fermat equation","hypergeometric motives","modular method"],"score":0,"title":"The generalized Fermat equation $x^3+y^3=z^n$","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Manvir Jaswal"],"reviews":0,"subject":"math.NT","summary":"We prove that the equations $x^3+y^3=z^n$ and $x^3+y^3=3z^n$ have no solution in coprime nonzero integers for any $n\\ge3$. For the first equation the open cases were prime exponents $p\\equiv1\\pmod 3$ above $10^9$ outside a set of congruence classes treated by Chen and Siksek; in these cases the solution $1^3+2^3=3^2$ blocks the modular method. We replace Kraus's Frey curve by a hypergeometric motive of rank three over $K=\\mathbb{Q}(\\sqrt{-3})$, whose parameter at a solution is $3$-adically close enough to a point of maximally unipotent monodromy for inertia at $\\sqrt{-3}$ to act unipotently;…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Opus","SageMath","PARI/GP","GAP","Python (python-flint/Arb, SymPy, mpmath)"],"discussed":0,"published":"2026-10-07","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"none"}