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

Juncheng Zhou, Hongfeng Wu

TL;DR

The paper proves that only finitely many finite fields have the property that every element is a sum of a $5$-potent and an $n$-potent. By case analysis on $(q-1)/(n-1)$ and Weil-type character sum bounds, it derives a universal bound $M=2809$ for the nonexistence of such representations in the $d\in\{2,3,4\}$ cases, and rules out $d=5$ by a simple counting argument; completing the claim requires Cohen et al.'s explicit computations for $q\le10000$. It then establishes a general finiteness theorem for fixed $m$, showing there exists an $M$ (with a computable bound $M=(2^mm)^2$) beyond which not every field element can be written as a sum of an $m$-potent and an $n$-potent, and extends these ideas to arbitrary subsets $A\subseteq\mathbb{F}_q$. The results yield corollaries for multi-potent sums, and the paper discusses algorithmic verification and open questions about tightening the bounds using the internal algebraic structure of the potent sets.

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 et al., we confirm 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.

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

TL;DR

The paper proves that only finitely many finite fields have the property that every element is a sum of a -potent and an -potent. By case analysis on and Weil-type character sum bounds, it derives a universal bound for the nonexistence of such representations in the cases, and rules out by a simple counting argument; completing the claim requires Cohen et al.'s explicit computations for . It then establishes a general finiteness theorem for fixed , showing there exists an (with a computable bound ) beyond which not every field element can be written as a sum of an -potent and an -potent, and extends these ideas to arbitrary subsets . The results yield corollaries for multi-potent sums, and the paper discusses algorithmic verification and open questions about tightening the bounds using the internal algebraic structure of the potent sets.

Abstract

We show that there are only finitely many finite fields whose members are the sum of an -potent element and a -potent element. Combining this with the algorithmic results provided by S.D. Cohen et al., we confirm 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.

Paper Structure

This paper contains 10 sections, 7 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Proof of Conjecture 1
  3. d=5
  4. d=2
  5. d=3
  6. d=4
  7. Completion of proof
  8. Some results on General problem
  9. Further problem
  10. Pari-gp code(given by S.D. Cohen et al.)

Key Result

Theorem 2.1

Suppose $q$ is a prime power and $n$ is a positive integer such that $n<q$ and $(n-1)\mid (q-1)$. Then, every element of $\mathbb{F}_q$ is the sum of a tripotent(=$3$-potent) and an $n$-potent if and only if $n=(q-1)/2$ and $q\in\{3,5,7,9\}$.

Theorems & Definitions (11)

  • Conjecture 1
  • Theorem 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 1 more