Giant Leap by CRM-ISM Postdoctoral Fellow

James Maynard, recipient of a 2013 CRM-ISM Postdoctoral Fellowship, has just announced a spectacular result in number theory which represents enormous progress in the quest to prove the Twin Primes Conjecture. This longstanding conjecture postulates that that there are infinitely many pairs of prime numbers which differ by 2 (as do, for example, 5 and 7). In April 2013, Yitang Zhang (University of New Hampshire) caused a sensation in the mathematical world by proving a weaker version of this conjecture, namely the existence of a finite bound B such that there are infinitely many pairs of distinct primes which differ by no more than B. Zhang proved the result with B = 70,000,000, while the original conjecture corresponds to B = 2, so the race was on to reduce the gap.

In November, recent Oxford PhD James Maynard, now working at the University of Montreal under the supervision of CICMA member Andrew Granville, announced that he had dramatically slashed this bound to 600, by a substantially easier method. Maynard is in good company: Terry Tao (UCLA), a recipient of the most presitigious prize in mathematics, the Fields Medal, had independently developed the same idea. Both were able to extend their proofs to show a much stronger result: given a number m, there corresponds a bound B such that there are infinitely many intervals (stretches of consecutive integers) of length B containing at least m distinct primes. If the ideas of Zhang, Maynard and Tao are pushed to their limit one could conceivably arrive at a bound (for 2 primes) as low as B = 12, close but not quite enough for the Twin Primes Conjecture.

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Source: Centre de recherches mathématiques.