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Is the Goldbach conjecture any easier if we allow primes to be negative as well? That is, every even integer is the sum or difference of two primes. The twin prime conjecture talks about the occurrence of a certain kind of prime gap, but I don't know if all even integers can be a prime gap (which would imply the Goldbach conjecture for $\Bbb Z$).

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  • $\begingroup$ Nobody else knows either. You might look up Polignac's conjecture. $\endgroup$ – Robert Israel Aug 14 '16 at 6:10
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    $\begingroup$ See also this MathOverflow discussion $\endgroup$ – Robert Israel Aug 14 '16 at 6:11
  • $\begingroup$ @RobertIsrael My conjecture is a priori a bit weaker than Polignac's, because I only want one prime gap for each even integer, not infinitely many. Unless there is an easy way to get infinitely many given a method producing one? $\endgroup$ – Mario Carneiro Aug 14 '16 at 6:13
  • $\begingroup$ Is it true that in some probabilistic model of the primes, the probability that $2n$, $n>0$ is a prime gap at least once is $1$? $\endgroup$ – Mario Carneiro Aug 14 '16 at 6:31
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    $\begingroup$ @Charles You should make that an answer, since I don't expect much more as this is an open problem. $\endgroup$ – Mario Carneiro Aug 14 '16 at 18:08
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Generally all the methods we have to produce gaps give us infinitely many, so the consensus seems to be that "infinitely many" is not much, or at all, harder than the "at least one" case.

Under Cramer's model this holds with probability 1.

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    $\begingroup$ Could you sketch the argument for the probability 1 claim? $\endgroup$ – Mario Carneiro Aug 14 '16 at 18:38
  • $\begingroup$ Not from my phone. Maybe next week when I'm home. $\endgroup$ – Charles Aug 14 '16 at 18:58
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    $\begingroup$ I'm sure the internet can provide a discussion of how the twin prime conjecture holds with probability $1$ in the Cramer model. Generalizing the difference from $2$ to $2N$ won't change anything in that argument. $\endgroup$ – Greg Martin Aug 14 '16 at 19:11

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