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So, the number has descended down from 70 million to a mortal 600! Expectedly, there is a new Polymath initiative to improve this further and who knows, perhaps 2 indeed is that number. Well, we are talking about the new result by James Maynard on the asymptotic gap between prime numbers.

Just to recap, the problem statement is as follows: If p_n and p_{n+1} are the nth and (n+1)th prime numbers (e.g., p_1=2,p_2=3,p_3=5, \ldots). On cursory counting, the gap p_{n+1}-p_{n} appears to grow larger as n, but not necessarily, because there are these well known twin prime pairs such as \left(3756801695685 \times 2^{666669} \pm 1 \right). Will this gap of 2 stay good forever as n \to \infty? If not as low as 2, will that be something low enough? i.e.,  How small can G be, where

G=\lim_{n \to \infty} \inf \left(p_{n+1}-p_{n}\right).

Earlier this year, Zhang proved that G is no more than $70$ million. That in a way proved that, the prime gap is bounded as we move along the number line. A bunch of mathematicians including Terrance tao worked further (polymath project on this) and improved that gap to as a few thousands. The latest result from Maynard brings in an independent proof for G=600. Maynard also claims that if the Elliott–Halberstam conjecture (See this nice blog post on prime tuple theory this by Terry Tao) is indeed true, then, G=12. Stunning!

What is stated here is just one avatar of the prime tuple theorem. More general results are also being discussed within the community. Terrance Tao again has this nicely articulated and maintains a polymath page for us. As an onlooker, I am as excited as many others to see this progress.

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August 2017
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