The Quartic Residues Latin Square

TL;DR

This paper uncovers a surprising pattern in the distribution of quartic residues modulo primes congruent to 5 mod 8, demonstrating a Latin square structure in a related matrix and linking it to sums of residues.

Contribution

It introduces a novel matrix construction from quartic residues that forms a Latin square and shows its independence from the generator choice, connecting it to residue sums.

Findings

The matrix M forms a Latin square under certain conditions.

The matrix M is largely independent of the generator g.

The sum of quartic residues relates to entries of M.

Abstract

We establish an elementary, but rather striking pattern concerning the quartic residues of primes $p$ that are congruent to 5 modulo 8. Let $g$ be a generator of the multiplicative group of $\mathbb Z_p$ and let $M$ be the $4\times 4$ matrix whose $(i+1),(j+1)-$th entry is the number of elements $x$ of $\mathbb Z_p$ of the form $x\equiv g^k \pmod p$ where $k\equiv i \pmod 4$ and $\lfloor 4x/p \rfloor = j$, for $i,j=0,1,2,3$. We show that $M$ is a Latin square, provided the entries in the first row are distinct, and that $M$ is essentially independent of the choice of $g$. As an application, we prove that the sum in $\mathbb Z$ of the quartic residues is $\frac{p}5(M_{11}+2M_{12}+3M_{13}+4M_{14})$.