Finite fields whose members are the sum of a potent and a 5-potent

TL;DR

This paper proves that only finitely many finite fields have elements that are sums of an n-potent and a 5-potent, confirming a conjecture and extending to a broader class of conditions.

Contribution

It establishes finiteness results for finite fields with elements as sums of n-potent and 5-potent elements, confirming a conjecture and providing elementary results for a general problem.

Findings

Finitely many finite fields have elements as sums of n-potent and 5-potent.

Confirmed the conjecture in Cohen et al. for all such finite fields.

Extended results to a general problem showing finiteness under broader conditions.

Abstract

We show that there are only finitely many finite fields whose members are the sum of an $n$-potent element and a $5$-potent element. Combining this with the algorithmic results provided by S.D. Cohen {\it et al.}, we confirm in the affirmative the conjecture in \cite{Cohen} concerning all finite fields satisfying this condition. Furthermore, we obtain several elementary results for General problem, proving that the number of finite fields satisfying general condition is also finite.