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Jun 2, 2014 at 7:34 history edited Aaron Meyerowitz CC BY-SA 3.0
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Jun 2, 2014 at 7:28 comment added Aaron Meyerowitz There is indeed no proof that there are infinitely many twin primes, but it is widely believed that much much more than that is true.
Jun 2, 2014 at 4:50 comment added barak manos Thanks. Indeed, if there was a proof that there are infinitely many twin primes $p,p+2$ such that $p+1$ is a product of at most $4$ primes (or in fact, any constant number of primes), then proving my conjecture would be trivial, since we must use $p+1 \in A$ (as I mentioned in the example). But to the best of my knowledge, there is no proof even for the fact that there are infinitely many twin primes.
Jun 1, 2014 at 22:43 history edited Aaron Meyerowitz CC BY-SA 3.0
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Jun 1, 2014 at 22:38 history answered Aaron Meyerowitz CC BY-SA 3.0