Group-Theoretic Remarks on Goldbach’s Conjecture

Abstract

The famous strongly binary Goldbach’s conjecture asserts that every even number 2n ≥ 8 can always be expressible as a 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 ≥ 11 is a prime number.