An Abelian Loop for Non-Composites

TL;DR

The paper introduces an abelian loop structure on a set of 1 and odd primes, exploring its properties using number theory and proposing related conjectures.

Contribution

It defines a novel abelian loop on non-composite numbers and analyzes its properties through number theory, proposing new conjectures.

Findings

The loop is well-defined and abelian.

Properties of the loop relate to number theory theorems.

Conjectures about the loop's behavior are proposed.

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.