On the Non-Existence of Primitive Solutions to the Generalized Fermat Equation via Frey Curves and Modularity
Abstract
We provide a detailed framework for proving Beal's conjecture: if A^x + B^y = C^z, x, y, z > 2, gcd(A, B, C) = 1, then no primitive solution exists. The approach uses a Frey-Hellegouarch curve, full computation of invariants c4, c6, Delta, Mazur-style irreducibility arguments, level-lowering at all primes including 2, and dimension arguments for spaces of modular forms.
Fields of Study & Subjects
Algebraic Geometry and Number Theory
Advanced Algebra and Geometry
Analytic Number Theory Research
Publication details
- Journal / source
- Zenodo (CERN European Organization for Nuclear Research)
- Publication date
- 31 January 2026
- Document type
- Preprint