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For consecutive (th and th)primes and , the asymptotic gap has got a fresh renewal off late. The famous twin prime conjecture says , but that is still a conjecture and not a proof. Recently, Zhang proved that is a finite number and a number definitely not bigger than . It was hoped that one day, the mathematical community will find a number lower than this and perhaps even the holy grail mark of .
Now what? Within a span of a month or so, the established gap of 70 million has improved to a thousand odd number and is still on a path of decent. Still some distance to the ultimate mark of 2, but boy, does collaboration work? Ever since the now famous breakthrough from Zhang touched the broad light (I had scribed my little thoughts on that earlier!), Terrance Tao and his team steadily managed to improve the bound. Tao has already knitted a nice and detailed summary on the progress. As of last week, the proven gap was , but now the updated gap could be as small as (which is still under verification as per polymath8 project page). Let us hope that, they can go all he way to prove the twin prime conjecture, one day!
It is interesting to read, from Tao’s blog (which any day is a treasure trove of many topics, thankfully written with a wider audience in mind) the several connections they made, including that to Elliott–Halberstam conjecture, for improving this fascinating distance between prime successors.