Nonexistence of twenty-fourth power residue addition sets

TL;DR

This paper proves the nonexistence of 24-th power residue difference sets in finite fields of prime order, completing the classification for all n < 24, using computational and algebraic methods.

Contribution

It establishes the nonexistence of 24-th power residue difference sets, filling a gap in the classification for all n less than 24.

Findings

Confirmed nonexistence of 24-th power residue difference sets in F.

Extended previous results to the case n=24.

Used computational algebra to derive formulas for cyclotomic numbers.

Abstract

Let n > 1 be an integer, and let F denote a field of p elements for a prime p = 1 (mod n). By 2015, the question of existence or nonexistence of n-th power residue difference sets in F had been settled for all n < 24. We settle the case n = 24 by proving the nonexistence of 24-th power residue difference sets in F. We also prove the nonexistence of qualified 24-th power residue difference sets in F. The proofs make use of a Mathematica program which computes formulas for the cyclotomic numbers of order 24 in terms of parameters occurring in quadratic partitions of p.