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.

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    $\begingroup$ Both are most probably true and thus logically equivalent, but I doubt that any of them implies another a priori. $\endgroup$ Dec 30 '14 at 11:24
  • $\begingroup$ There is a small section about these two conjectures in "Gödel, Escher Bach ..." by Hofstadter, (a book every mathematician should read, or at least know about, IMHO. $\endgroup$ Dec 30 '14 at 14:06
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    $\begingroup$ @PerAlexandersson Thanks for the reference; I'll look it up. I've held out on GEB thus far due to low reviews from people actually working in these fields -- my impression is that it's a popular-science book with the shortcomings typical of that genre. $\endgroup$
    – Newb
    Dec 30 '14 at 14:23
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    $\begingroup$ If assuming one conjecture ever happened to help proving the other, it stands to reason that that one would be Goldbach's. (Notice that every odd number is the difference of 2 squares, but very few are the sum of 2 squares.) $\endgroup$ Dec 30 '14 at 15:50
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    $\begingroup$ I strongly second the observation that "Goedel-Escher-Bach" is most popular among people who know little about any of the things involved. It may be a fun read while in that state, but quickly becomes irritating if one knows about the stuff, especially if one has an idea about two or more of those topics. Indeed, much of the "analogy" is fake/forced, in my opinion... $\endgroup$ Dec 30 '14 at 19:43

I don't think they are equivalent, since it is conjectured that every even number is the difference of two consecutive primes infinitely often, while in Goldbach the number of solutions is finite.

This is Polignac's conjecture.

Another major difference: If Goldbach's conjecture were false, it could have been disproved with finite computation -- enumerate the primes to $n/2$ and this doesn't work for difference of two primes.


Close to nothing can be said rigorously about your question, but I believe the following heuristics:

  • It's hard to imagine the exact solution of either conjecture not leading to substantial progress in the resolution of the other. This is because the current methods that handle structural results about primes have similar limitations for both problems. See Terence Tao's blog.
  • There is a sense in which the two problems have different flavor, if you look at their "approximate" versions. We can show results about differences of primes falling in certain bounded intervals, while no analogous result is known about sums of primes. See Terence Tao's blog.

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