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Very recently, Yitang "Tom" Zhang from the University of New Hampshire made a huge step toward proving the twin prime conjecture. He proved that there are infinitely many pairs of prime separated by a constant distance, and that distance is less that $7 \cdot 10^7$. For example, in the special case of the twin prime conjecture, pairs of primes are separated by 2, such as 3 and 5, 5 and 7, 11 and 13, etc. Now people are saying that they might be able to use his methods to prove that the difference is lower than 16. My question is, does his proof limit the possibility of these pairs of primes to only one pair, or does it potentially allow for all the 7 million possibilities? Thanks! By the way this is where I found out about this: https://www.youtube.com/watch?v=vkMXdShDdtY

vonbrand
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Ovi
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  • The main theorem is that there are infinitely many integers $n$ for which the interval $[n,n+7\times10^7]$ contains at least two primes. I'm not sure what you mean by "limit...to one pair". Certainly there are at most $35$ million possibilities, not $70$ million. – Erick Wong May 28 '13 at 19:18
  • Here's the paper by Yitang Zhang on Gap between primes. And also there was a question asked on Math.SE before. The actual theorem says that the gaps between two primes tending to infinity is less than $70$ million. – Inceptio May 28 '13 at 19:30

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The proof does not give any specific $k$ for which $(n,n+k)$ is prime infinitely often. But such a $k\le7\cdot10^7$ does exist by his theorem. In fact the theorem even implies that there is a $k$ such that there are infinitely many consecutive primes $(n,n+k).$

Charles
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  • I understand that but does it allow for multiple pairs of these primes, such as p, p+k, n, n+r, a, a+c, or does it just restrict it to one possible n, n+k? – Ovi May 28 '13 at 19:37
  • @Ovi: It does not prove that there is more than one admissible $k$. Of course it doesn't show there aren't others... – Charles May 28 '13 at 19:46