Close encounters among the primes

This paper was written, apart from one technical correction, in July and August of 2013. The, then very recent, breakthrough of Y. Zhang [18] had revived in us an intention to produce a second edition of our book “Opera de Cribro”, one which would include an account of Zhang’s result, stressing the sieve aspects of the method. A complete and connected version of the proof, in our style but not intended for journal publication, seemed a natural first step in this project. Following the further spectacular advance given by J. Maynard (arXiv:1311. 4600, Nov 20, 2013), we have had to re-think our position. Maynard’s method, at least in its current form, proceeds from GPY in quite a different direction than does Zhang’s, and achieves numerically superior results. Consequently, although Zhang’s contribution to the distribution of primes in arithmetic progressions certainly retains its importance, the fact remains that much of the material in this paper would no longer appear in a new edition of our book. Because this paper contains some innovations that we do not wish to become lost, we have decided to make the work publicly available in its original format. We are extremely pleased to be able to include it in the current volume, commemorating the one hundred and twenty-fifth anninversary of the birth of Srinivasa Ramanujan.