{"id":"hexagon/2610.00153","pdf":"https://hexagonmath.org/pdf/2610.00153v1","url":"https://hexagonmath.org/abs/2610.00153v1","tags":["msc:11D41","msc:11R29","msc:11Y40","msc:11Y50"],"score":0,"title":"The generalized Fermat equation $x^2+y^5=z^7$","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Manvir Jaswal"],"reviews":0,"subject":"math.NT","summary":"We show that the equation $x^2+y^5=z^7$ has no solution in nonzero coprime integers, assuming a classification theorem from Putz's thesis, which has not appeared in a refereed journal. Equivalently, none of the equations $x^2+y^5=z^7$, $x^2+y^7=z^5$, $x^5+y^7=z^2$ has a solution in nonzero coprime integers. Putz attaches to a solution an octic algebra, the fibre of a Belyi map of degree $8$, and proves that it is always isomorphic to one fixed octic field $L_8$. We prove unconditionally that no solution has fibre isomorphic to $L_8$, by a fifth-power descent over a field $L_{24}$ of degree…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Opus","PARI/GP","Python (python-flint/Arb, NumPy, SciPy, SymPy, mpmath)"],"discussed":0,"published":"2026-10-08","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":true,"formalizations":["https://palomar-registry.org/entry?id=PALOMAR-2026-10-09-000002&version=1"],"human_understanding":"none"}