# Group-theoretic remarks on Goldbach's conjecture

## TL;DR

This paper introduces a novel group-theoretic approach using element order prime graphs of alternating groups to analyze Goldbach's conjecture, proving it for specific cases related to primes greater than or equal to 11.

## Contribution

It presents six new group-theoretic formulations of Goldbach's conjecture and proves its validity for numbers related to primes p ≥ 11.

## Key findings

- Proved Goldbach's conjecture for numbers p+1 and p-1 when p ≥ 11 is prime.
- Established six group-theoretic equivalents of Goldbach's conjecture.
- Applied element order prime graphs of alternating groups to number theory problems.

## Abstract

The famous strongly binary Goldbach's conjecture asserts that every even number $2n \geq 8$ can always be expressible as the sum of two distinct odd prime numbers. We use a new approach to dealing with this conjecture. Specifically, we apply the element order prime graphs of alternating groups of degrees $2n$ and $2n-1$ to characterize this conjecture, and present its six group-theoretic versions; and further prove that this conjecture is true for $p+1$ and $p-1$ whenever $p \geq 11$ is a prime number.