  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.
