One open conjecture is that every even integer greater than two is the *difference* of two primes. (Some superficial discussion here.)

Goldbach's conjecture states that every even integer greater than two is the *sum* of two primes.

The big question: are the two equivalent? That is to say, do these conjectures imply each other? I spent a bit of time pursuing this question, and I did not find a satisfactory answer.

I now suspect that the two are actually not equivalent -- if they were, then I think it would suggest a symmetry on the prime numbers that I don't think they have.

Anyway, I'd be glad to hear your input on the matter.