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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.

“Bounded gaps between primes” by Yitang Zhang now available

To complete the story started as a rumour report in ‘Primes gotta stick together‘ and confirmed in ‘Primes really do stick together‘, here we report that Annals of Mathematics has posted the PDF of ‘Bounded gaps between primes‘ by Yitang Zhang on its ‘to appear in forthcoming issues’ page. After the seminar on 13th May, Zhang apparently submitted a revised manuscript on 16 May, which was accepted 21 May 2013. So if you’ve been itching for details, here’s your chance (assuming you have access to a subscription to Annals).

Here’s the abstract:

It is proved that \[ \liminf_{n\to \infty}\, (p_{n+1} – p_n) < 7 \times 10^7 \text{,}\] where $p_n$ is the $n$-th prime.

Our method is a refinement of the recent work of Goldston, Pintz and Yildirim on the small gaps between consecutive primes. A major ingredient of the proof is a stronger version of the Bombieri-Vinogradov theorem that is applicable when the moduli are free from large prime divisors only (see Theorem 2), but it is adequate for our purpose.

The paper: Bounded gaps between primes by Yitang Zhang, in Annals of Mathematics.

All odd integers greater than 7 are the sum of three odd primes!

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+

Primes really do stick together


“The author has succeeded to prove a landmark theorem in the distribution of prime numbers. … We are very happy to strongly recommend acceptance of the paper for publication in the Annals.”

According to the Nature News blog, at yesterday’s seminar given by Yitang Zhang it was revealed that his proof that there are infinitely many pairs of primes less than seventy million apart has already been refereed for the Annals of Mathematics; that’s a quote from the referee’s report above.

It seems the proof doesn’t use any unconventional machinery (in contrast to Mochizuki’s Proof from Planet 9 of the abc conjecture) and is fairly uncontroversial. How pleasant! Of course, someone might find a problem with it once it’s publicly available, but that’s the way for all things.

Source: First proof that infinitely many prime numbers come in pairs at Nature News

ABC, as easy as pp1-40

Here’s something that slipped to the bottom of our news queue: Shin Mochizuki has uploaded a 40-page overview of his “Inter-universal Teichmüller theory” papers – the ones which he claims prove the abc conjecture.

Don’t expect to understand any of it, but maybe someone else will.

PDF: A Panoramic Overview of Inter-universal Teichmüller Theory by Shinichi Mochizuki

Previously: Proof News

via Jordan Ellenberg on Twitter.

Proof News

Here’s a little catch-up with the status of the claimed proofs of some big statements that were announced recently.

At the end of August, Shin Mochizuki released what he claims is a proof of the abc conjecture (link goes to a PDF). Barring someone spotting a huge error, it’s going to take a long time to verify. It’s mainly quiet at the moment, apart from a claimed set of counterexamples to one of Mochizuki’s intermediate theorems posted by Vesselin Dimitrov on MathOverflow, which was quickly shut down because the community there didn’t approve of MO being used to debate the validity of the proof. No doubt there are other niggles being worked out in private as well.

At the start of September, Justin Moore uploaded to the arXiv what he claimed was a proof that Thompson’s group F is amenable. Like Mochizuki’s abc proof, experts thought Moore’s proof was highly credible. We were waiting for my chum Nathan to write about it, since his PhD was all about Thompson’s groups F and V, but it turns out we don’t need to: at the start of this week, Justin retracted his paper because of an error which “appears to be both serious and irreparable”. The amenability of Thompson’s group F has been proven and disproven many times, so I still want Nathan to tell me (and you) all about it.

In lighter news, via Richard Green on Google+, recent uploads to the arXiv show that Goldbach’s conjecture and the Riemann hypothesis are true. I’d love to know how it feels to upload a six-page paper which you know proves something like the Riemann hypothesis. It must be a lovely state of mind. Certainly much better than what people like Moore and Mochizuki must go through, waiting for the first email to arrive telling them they’ve made a terrible mistake and their work is not yet complete.

If I’ve inspired you to have a go yourself, look at Wikipedia’s list of unsolved problems in mathematics and take a crack at one this weekend. Can’t hurt1 to try!

  1. Disclaimer: depending on levels of ability, perseverance and agreement with consensus reality, attempts to solve these problems may well ruin your life []