Just wondering if anyone knows: Do all of these results, from the original Zhang constant down to Maynard's 600, imply only that there exists *at least one* number N no more than the constant such that there are infinitely many primes separated by exactly N ??? Or is there some reasoning within the proofs of these theorems that would imply more than one such N ??? --Dan On 2013-11-20, at 6:46 AM, Fred Lunnon wrote:
Since this topic has subsided from public view, a priesthood incorporating James Maynard, Andrew Granville, Terry Tao, has been assiduousy shaving Zhang's original 70 million, and refining and simplifying the proof to the point where a normal mathematically-inclined human (if such an entity exists) might possibly be able to follow it with the assistance of a powerful computer and a sabbatical year or two --- by which time it would no doubt have undergone further drastic improvement.
The current result asserts that there are infinitely many consecutive prime pairs with difference h for some h <= 600 . The result generalises to finite arithmetic progressions of m consecutive primes with span h , for which instead h <= exp(8m + 5) .
For those with time to spare and robust self-esteem, there is (considerably) more detail available at http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-m...
Fred Lunnon