On existence questions for the functional equations $f^8+g^8+h^8=1$ and $f^6+g^6+h^6=1$

TL;DR

This paper extends the non-existence results for non-constant solutions of the functional equations involving sums of powers equal to one, specifically for the cases n=8 with meromorphic functions and n=6 with entire functions.

Contribution

It proves the non-existence of non-constant meromorphic solutions for $f^8+g^8+h^8=1$ and entire solutions for $f^6+g^6+h^6=1$, filling gaps in previous research.

Findings

No non-constant meromorphic solutions for $f^8+g^8+h^8=1$.

No non-constant entire solutions for $f^6+g^6+h^6=1$.

Answers to open questions posed by G. G. Gundersen.

Abstract

In 1985, W.K.Hayman (Bayer. Akad. Wiss. Math.-Natur. Kl. Sitzungsber, 1984(1985), 1-13.) proved that there do not exist non-constant meromorphic functions $f,$ $g$ and $h$ satisfying the functional equation $f^n+g^n+h^n=1$ for $n\geq 9.$ We prove that there do not exist non-constant meromorphic solutions $f,$ $g,$ $h$ satisfying the functional equation $f^8+g^8+h^8=1.$ In 1971, N. Toda (T\^{o}hoku Math. J. 23(1971), no. 2, 289-299.) proved that there do not exist non-constant entire functions $f,$ $g,$ $h$ satisfying $f^n+g^n+h^n=1$ for $n\geq 7.$ We prove that there do not exist non-constant entire functions $f,$ $g,$ $h$ satisfying the functional equation $f^6+g^6+h^6=1.$ Our results answer questions of G. G. Gundersen.