The Generalized Fermat Equation x² + y³ = z²⁵

2025/06/12 by Nuno Freitas, Michael Stoll, Freitas, Nuno +1
Computer Science · Mathematics · #11G30 #11G35 #14K20 #Algebraic Geometry and Number Theory #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2506.10667

openalex publication_date 2025/06/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the generalized Fermat equation (*) x² + y³ = z²⁵. Using the known parameterization of the primitive integral solutions to x² + y³ = z⁵ (due to Edwards), we reduce the solution of (*) to the solution of five specific equations of the form H(u,v) = w⁵, where H is homogeneous of degree 10 with coefficients in a sextic number field K, u and v are coprime (rational) integers, and w ∈ K.

Citations

• On Galois inertial types of elliptic curves over ℚ_ℓ
• The generalized Fermat equation with exponents 2, 3, n
• On the symplectic type of isomorphims of the p-torsion of elliptic\n curves

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