On Fermat Diophantine functional equations and little Picard theorem

TL;DR

This paper explores conditions under which certain Fermat-type functional equations lack non-trivial meromorphic solutions, offering new perspectives on proofs related to the little Picard theorem.

Contribution

It introduces equivalence conditions for the non-existence of solutions to Fermat Diophantine functional equations, providing alternative approaches to the little Picard theorem.

Findings

Identifies conditions for non-existence of meromorphic solutions

Provides new approaches to prove little Picard theorem

Links Diophantine equations with complex analysis

Abstract

We discuss equivalence conditions on the non-existence of non-trivial meromorphic solution to the Fermat Diophantine equations $f^m(z)+g^n(z)=1$ with integers $m,n\geq2$, from which other approaches to prove little Picard theorem are provided.