Deriving from an Assumption a semistable elliptic Curve for \(x_{0}^{k} + y_{0}^{l} = z_{0}^{m}\)

TL;DR

This paper demonstrates that the generalized Fermat equation has no coprime integer solutions by deriving an elliptic curve from the assumption of solutions and applying prior results by Frey, Ribet, and Wiles.

Contribution

It shows how to derive a semistable elliptic curve from the generalized Fermat equation to prove its unsolvability in coprime integers.

Findings

No coprime integer solutions exist for the generalized Fermat equation.

The derivation of an elliptic curve from the equation supports the proof of unsolvability.

The approach leverages prior results by Frey, Ribet, and Wiles.

Abstract

With a simple transformation of the three exponents the generalized Fermat equation can be put into the same form as the Fermat equation. When it is rewritten into this new altered form any real solutions to the altered equation equal a subset of real solutions to the Fermat equation. This result, along with a paradigm previously established by G. Frey and Y. Hellegouarch, can be used to show that the generalized Fermat equation has no solutions in coprime integers. In fact we prove one can derive an elliptic curve over $\mathbb{Q}$ for primes (p \geq 3), from the assumption that (x_{0}^{k} + y_{0}^{l} = z_{0}^{m}), (k, l, m \geq 3), has coprime solutions in integers. The curve then can be used to show the unsolvability of this equation. Our aim is not to arrive at very deep results. Rather the goal in this paper is to apply prior results obtained by G. Frey, K. Ribet and A. Wiles, to show the lack of solutions to the generalized Fermat equation.