Terence Tao has just uploaded a preprint to the arXiv with a claimed proof of the Erdős discrepancy problem.
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The big news last year was the quest to find a lower bound for the gap between pairs of large primes, started by Yitang Zhang and carried on chiefly by Terry Tao and the fresh-faced James Maynard.
Now that progress on the twin prime conjecture has slowed down, they’ve both turned their attentions toward the opposite question: what’s the biggest gap between subsequent small primes?
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.
Summer is a busy time for this site’s hard-working triumvirate, so we haven’t been keeping on top of the news as much as we’d like. There’s been some quite interesting news, so here’s a quick round-up of the most important bits:
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.
It seems that big mathematical advances are like buses – you wait ages for one, and then two come along at once. Also revealed yesterday was a proof of the odd Goldbach conjecture: that all odd numbers greater than 7 can be written as the sum of exactly three odd primes. The proof is contained in Major arcs for Goldbach’s theorem, a paper submitted to the arXiv by Harald Helfgott, who’s a mathematician at the École Normale Supérieure in Paris. This new paper completes the work started in Helfgott’s previous paper, Minor arcs for Golbach’s problem, published last year.
The strong Goldbach conjecture states that every even number can be written as the sum of two primes. This is still unproven, and remains one of the long-standing unproven results in number theory. Sadly, it’s the opinion of Terence Tao, among others, that the method used to prove the weak conjecture probably won’t work on the strong conjecture.
The paper: Major arcs for Goldbach’s theorem by Harald Helfgott
via Terry Tao on Google+
I’m hijacking Katie’s newly-instituted series of posts about who to follow on Twitter with a post about who to follow on Google+.
Google+ famously has almost nobody on it. If anyone knows the potential for really interesting exceptions to the word “almost”, it’s mathematicians, so by that mad logic there should be some really interesting mathematicians on Google+. As luck has it, there are! It seems that the unconstrained nature of Google+ posts gives mathematicians the space they need to express themselves usefully.
Here are a few mathsy people you might like to encircle on Google+.