An Abelian Loop for Non-Composites

Abstract

We define an abelian loop on a set $S$ consisting of 1 and all odd prime numbers with an operation $\bullet$, where for $a,b$ $\in$ $S$, $a$ $\bullet$ $b$ is the smallest element of $S$ strictly larger than $|a-b|$. We use theorems and conjectures from number theory to prove properties of the loop and state analogous conjectures about the loop.