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Terence Tao has solved the Erdős discrepancy problem!

Terence Tao has just uploaded a preprint to the arXiv with a claimed proof of the Erdős discrepancy problem.

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Prime gaps update

There’s been some progress on the bounded gaps between primes front since we last checked in.

The Polymath8 project has got the gap down to $4,680$. But that’s small beans: James Maynard, a postgrad student at Oxford, announced at a meeting in Oberwolfach that he has got the gap down to $700$. Emmanuel Kowalski has written an effusive post on his blog singing the praises of Maynard’s achievement.

Bound on prime gaps bound decreasing by leaps and bounds

Update 17/06/2013: The gap is down to 60,744. That’s a whole order of magnitude down from where it started!

When Yitang Zhang unexpectedly announced a proof that that there are infinitely many pairs of primes less than 70 million apart from each other – a step on the way to the twin primes conjecture – certain internet wags amused themselves and a minority of others with the question, “is it a bigger jump from infinity to 70 million, or from 70 million to 2?”.

Of course the answer is that it’s a really short distance from 70 million to 2, and here’s my evidence: the bound of 70 million has in just over three weeks been reduced to just a shade over 100,000.

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