The generalized Fermat equation $x^2+y^5=z^7$

Manvir Jaswal

The generalized Fermat equation $x^2+y^5=z^7$

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Contributed by Manvir Jaswal ↗

Version 1 / Oct 08, 2026 / CC BY 4.0

Abstract

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

24

that Putz introduced, completed by fifth-power residue symbols at five auxiliary primes. We also prove, without assuming the generalized Riemann hypothesis, that

L_{24}

has class number

1

, using a criterion of Belabas, Diaz y Diaz and Friedman.

Provenance statement

The author designed and operated the multi-agent research system used in this work, built on Anthropic Claude Opus 5.5. The system carried out literature analysis, mathematical development and computation, with checks assigned to agents separate from those developing the arguments. Claude Opus 5.5 was used to draft and revise the manuscript. Programs and data: https://doi.org/10.5281/zenodo.22983849. Lean formalization of Theorems 1.1 and 1.2: https://doi.org/10.5281/zenodo.23065698.

FormalizationsPalomar

Tools used

Anthropic: Claude OpusVersion 5.5
PARI Group: PARI/GPVersion 2.17.3
Python Software Foundation: Python (python-flint/Arb, NumPy, SciPy, SymPy, mpmath)Version 3.12

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Version history

v1Submitted by Manvir JaswalInitial depositCurrentOct 07, 2026