There’s a new paper out on the arXiv last night, Small gaps between primes, by James Maynard, which brings the bound on the size of gaps between primes down to 600. This uses some new methods, beating out the Polymath8 project, which has been improving Zhang’s original bound of 70,000,000, getting it down to 4680.

To follow the Polymath8 project, the place to look is Terence Tao’s blog, here. They’re working on a paper, with the current draft version available here. This is a remarkable collaborative project bringing together a sizable group of mathematicians in an unusual way.

For more about this, see this expository article by Andrew Granville, which is pre-Maynard. At Quanta magazine, Erica Klarreich has an excellent long popular article telling the story to date, including that of Maynard’s new result.

Interesting to know if Zhang didn’t publish his twin primes work back in May, Maynard’s paper would still show as it is now 6 months later, would it be an even BIGGER twin primes breakthrough as well just by now?

Thanks, Peter, for the update. I’ve had a number of things going on in my life at this time, and I haven’t had the opportunity to stay abreast of the latest developments in the twin prime conjecture. Even though number theory isn’t my primary interest in mathematics, every mathematician has a soft spot in his/her heart for what Carl Friedrich Gauss described as the “Queen of Mathematics”.

Maybe more to the point, Maynard can not only show a better bound on lim inf gaps between one prime and the next (the next big excitement will be when that gets to single digits, we’re still over 100, and in some sense that’s not so much nicer than Zhang’s 70,000,000) but he can show that there is a finite bound on lim inf gaps over any given m primes – this wasn’t thought to be on the table. Roughly, we knew bounded gaps between primes conditional on a (still) unproved `good behaviour’ conjecture (Elliott-Halberstam); Zhang’s theorem is powered by Zhang’s observation that you don’t need the full strength of that conjecture, and you can actually prove (with a lot of work) what you do need. But even assuming the full E-H conjecture we couldn’t get much below trivial lim inf gaps containing multiple primes before now (Goldston and Yildirim claimed something like this ten years ago, but there was an error that looked unfixeable: Maynard, roughly speaking, fixed it).

Andrew Granville has updated his expository article to include Maynard’s results. See http://www.dms.umontreal.ca/~andrew/CEBBrochureFinal.pdf.

@TonyK

Thanks for the link. Makes good reading on a cold stormy afternoon.

@PW

Fascinating stuff even for an old experimental physicist.

On this topic of abstract science like pure mathematics, are you planning to see “Not Even Wrong”, an artistic production of dance at Geneva in May 2014? It seems to be based on some ideas in your book:

http://kylie-walters.com/?portfolio=not-even-wrong-n-e-w

http://www.youtube.com/watch?v=xMSyHTLFwM0

http://www.plateaux.ch/en/show/not-even-wrong-new/

a,

Thanks. That’s news to me. But no plan at the moment to be in Europe then.

Does anyone know if there has been any progress in confirming the accuracy of Harald Helfgott’s proof of the weak Goldbach’s convention as he claimed in a pre-print last May?

(This would imply that all odd numbers are the sum of not more than three odd primes and that all even numbers are the sum of not more than four odd primes, subject to trivial exceptions under 7.)

I bring it up now because it was announced the same day as the 70,000,000 apart prime result of Zhang.

On a lighter note, remember Tom Lehrer’s Lobachevsky song?

http://www.sing365.com/music/lyric.nsf/Lobachevsky-lyrics-Tom-Lehrer/D97B21BF6516390448256A7D0024B8B9