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But now, as the Mathematician Zhang Yitang from University of New Hampshire in Durham has shown, there is a kind of weak version of the twin prime conjecture. He didn’t prove that a distance of 2 exists for an infinite number of primes, but he did prove that there are infinitely many prime gaps shorter than 70 million.

~ Mathematician Makes Astounding Prime Numbers Breakthrough

Assuming this proof holds up, what would the implications of this discovery be? Are there any branches of engineering or science that are likely to be able to benefit from this knowledge, or is this likely - for now at least - only to impact the work of pure mathematicians?

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One important question that people often forget to ask is whether the methods used to prove the result have implications in other areas (e.g. Fermat's Last Theorem). I don't know whether that is the case here, but knowing a way to solve a difficult problem is sometimes more useful than knowing the solution. –  Matt Pressland May 14 '13 at 16:17
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The twin-prime conjecture is already assumed to be true by many. A weaker version (with $\le 16$ in place of $2$) has been proved assuming the Elliott–Halberstam conjecture, in 2004 by Daniel Goldston, János Pintz and Cem Yıldırım. Their result is much stronger than this one. The interesting thing about this one is that it doesn't require any unproved conjectures, so it's a big breakthrough and its methods would likely lead to new things, but it's premature to ask for its applications to engineering. –  ShreevatsaR May 15 '13 at 5:21
    
Solving these conjectures gets us closer to understanding the behavior of the primes themselves. One of the endgoals for understanding prime numbers (and certainly not the only one) is to be able to exactly predict them as well as understand fully their behavior. The twin prime conjecture is just a very small step towards that goal which would have a profound impact on cryptography as well as more generally any sort of processes involving integer data should the goal be accomplished –  frogeyedpeas Jul 1 '13 at 2:25
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closed as not constructive by Thomas Andrews, Lord_Farin, ShreevatsaR, Zander, J. M. May 18 '13 at 6:10

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