Representation of the elements of the finite field ${\mathbb {F}}_p$ by fractions

Abstract

We prove the apparently unnoticed observation that the p elements of the finite field ${\mathbb {F}}_p$, where $p\ge 3$ is a prime, can be represented by fractions with numerators and denominators less than $\sqrt{p}$. Notice that there are asymptotically only cp such fractions, where $c=\frac{12}{\pi ^{2}} =1.21585\ldots$.