The Quartic Residues Latin Square

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 $x\equiv g^k(\mathrm{mod}p)$ where $k\equiv i(\mathrm{mod}4)$ 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 $\frac{p}{5}(M_{11}+2M_{12}+3M_{13}+4M_{14})$.