{"id":180,"date":"2013-06-14T02:12:43","date_gmt":"2013-06-14T07:12:43","guid":{"rendered":"http:\/\/blogs.ams.org\/blogonmathblogs\/?p=180"},"modified":"2013-06-14T09:35:05","modified_gmt":"2013-06-14T14:35:05","slug":"narrowing-the-gap","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/blogonmathblogs\/2013\/06\/14\/narrowing-the-gap\/","title":{"rendered":"Narrowing the Gap"},"content":{"rendered":"

A group of seasoned mathematicians including Terence Tao are working to improve on Yitang Zhang<\/a>\u2019s exciting recent proof that there are infinitely many pairs of consecutive primes separated by no more than 70 million.\u00a0 A glance at the polymath8 wiki<\/a> shows that the gap has now been shrunk to a measly 250,000.\u00a0 Although this is still nowhere near the gap of two required by the twin primes conjecture, one fact stands out to me as a junior faculty. \u00a0As Mario Livio recently stated in his new book<\/a> \u201cThe road to triumph [is] paved with blunders.\u201d And on the path to furthering their (and our) knowledge, those contributing to polymath8 are recording all progress, including mistakes! \u00a0It’s exciting to see research happening in real time, and even as I wrote this, I noticed new World Records being established for the smallest bound on the gap.\u00a0 The current record-holder is MIT Computational Number Theorist Andrew Sutherland<\/a>.<\/p>\n

Terrence Tao\u2019s June 3rd<\/sup> post<\/a>\u00a0gives a great introduction to recent events as\u00a0he clearly lays out the definition of H admissible k-tuples of integers and how they relate to the twin primes conjecture.\u00a0 \u00a0A k-tuple of integers (a list of k distinct integers) is admissible if it avoids at least one residue class modp for every prime p. These are the k-tuples that have a fighting chance at possessing the desirable characteristic that infinitely many translates consist exclusively of primes.\u00a0 \u00a0For instance, an example of a NON-admissible 3-tuple is {-2, 0, 2} since each residue class mod 3 is represented.\u00a0 Notice that every translate of\u00a0{-2,0,2} must contain a multiple of 3, and therefore there are only finitely many translates containing only primes.\u00a0 Filtering all k-tuples to find these special ones is accomplished by creating a \u201csieve\u201d, thus a series of posts on sieve theory can be found on Tao\u2019s blog and a quick encapsulation of many sieves can be found on another polymath wiki<\/a>.<\/p>\n

The famous Hardy-Littlewood conjecture asserts that all H-admissible k-tuples have infinitely many translates consisting exclusively of primes.\u00a0 But this over-arching theorem is extremely difficult (the twin primes conjecture is a corollary where k=2).\u00a0 Tao proceeds to explain various dilutions and progress made including the 2005 breakthrough by Goldston-Pintz-Yildirim.\u00a0 Apparently, finding narrow H-admissible k-tuples leads to a reduction in the gap between consecutive primes, thus improving on Zhang\u2019s result.\u00a0 This is just one approach to improving the result.<\/p>\n

As a non-number theorist, it\u2019s great to be able to understand at least the introductory notions used to explore these questions and to see some of the relationships between older and newer results.\u00a0 It seems that Zhang used many common techniques, but applied them in novel ways, and presented his work so clearly that it was quickly confirmed as valid. \u00a0While many articles have dwelled on Zhang’s relative obscurity, it\u2019s worth noting that Zhang clearly has a taste for \u00a0problems that are easy to state and hard to solve as he tackled The Jacobian Conjecture for his dissertation work. \u00a0Zhang\u2019s dissertation was in the field of Algebraic Geometry (not number theory), and that his advisor remembers him as \u201ca free spirit\u201d<\/a> who did not even request letters of recommendation for jobs! \u00a0 \u00a0From now on it seems that Zhang need only his own accomplishments to recommend him.<\/p>\n

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A group of seasoned mathematicians including Terence Tao are working to improve on Yitang Zhang\u2019s exciting recent proof that there are infinitely many pairs of consecutive primes separated by no more than 70 million.\u00a0 A glance at the polymath8 wiki … Continue reading →<\/span><\/a><\/p>\n

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