Suppose that we identify primes to be both positive and negative
integers, p, such that
there does not exist non-units j and k, such that j * k = p.
There could be some argument about whether or not to admit (-1) as a prime.
It is useful in some cases to include (-1) as a prime.
In either case,
The twin prime conjecture, "there exist infinitely many prime pairs such
that their difference is equal to 2",
becomes indistinguishable from
"there exist infinitely many prime pairs such that there sum is 2."
On the other hand,
Goldbach's conjecture, "For each even number, there exist some pair of
primes which add to that even number"
becomes indistinguishable from,
"For each even number, there exist some pair of primes such that their
difference is equal to that even number."
Thus both Goldbach's conjecture and the twin prime conjecture become
"For each even integer there exist infinitely many primes such that
their sum or difference is equal to that even number".