TITLE:
Group-Theoretic Remarks on Goldbach’s Conjecture
AUTHORS:
Liguo He, Gang Zhu
KEYWORDS:
Alternating Group, Element Order Prime Graph, Goldbach’s Conjecture, Centralizer
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.12 No.11,
November
9,
2022
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.