A special seminar has been scheduled for tomorrow (Monday) at 3pm at Harvard, where Yitang Zhang will present new results on “Bounded gaps between primes”. Evidently he has a proof that there exist infinitely many different pairs of primes p,q with p-q less than ~~17,000,000~~ 70,000,000.

Whether this proof is valid should become clear soon, but there still seems to be nothing happening in terms of others understanding Mochizuki’s claimed proof of the abc conjecture. For an excellent article describing the situation, see here.

**Update**: The “bounded gaps” talk is now on the Harvard seminar listing with abstract

The speaker proves that there are infinite number of pairs of primes whose difference is bounded by 70 million.

For more on the significance of this, see this Google+ posting by David Roberts.

I haven’t seen a paper, but rumor is that one exists and two referees at a major journal have found it to be correct.

**Update**: The most recent version of Mochizuki’s lecture notes for a general talk about his work is here. As mentioned in the Caroline Chen article, Go Yamashita has been talking to Mochizuki. Yamashita has now posted a short document FAQ on “Inter-Universality” and promises “For the details of the theory, please wait for the survey I will write in the near future.” He also notes:

I refuse all of the interviews from the mass media until the situation around the papers will be stabilised.

**Update**: In a weird coincidence, another major analytic number theory result is out today, a proof by Harald Helfgott of the ternary Goldbach conjecture. This says that every odd integer greater than 5 is the sum of three primes. The result had been known for all integers above e^{3100}, and Helfgott’s proof reduces that bound to 10^{30} which is small enough so that all smaller values can be checked by computer.

**Update**: *Nature* has a story up about the Zhang result, including details of one of the *Annals* referee reports (I gather the paper will be published there).

**Update**: For some background to the methods being used by Zhang, see here. For Terry Tao on Zhang, see here, on Helfgott, here.

**
Update**: New Scientist has a story about the Zhang result here, with quotes from Iwaniec, who has reviewed the paper, finding no error.

**Update**: A report from the talk at Harvard is here.

**Update**: There’s more about the Zhang proof at Emmanuel Kowalski’s blog, including a link to the Zhang paper.

**Update**: Nice piece about this in Slate from Jordan Ellenberg.

To Peter Woit, OK it’s your blog, and I enjoy reading it, so I’ll stop with that

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At the oh-so-british “The Register” (Motto: ‘Biting the Hand that feeds IT’) we read:

I know who ‘Satoshi Nakamoto’ is, says Ted Nelson

Sociologist, philosopher, computer industry pioneer and inventor of the term “hypertext” Ted Nelson is claiming that he knows the identity of Bitcoin inventor “Satoshi Nakamoto”. In a rambling – and, let’s face it, odd – 12-minute post on YouTube, Nelson spins out the suspense, throws in a dialogue with himself as Sherlock Holmes and Doctor Watson, and finally ends with the statement that the mystery developer of the cryptocurrency is Japanese mathematician Shinichi Mochizuki, research professor of mathematics at Kyoto University. … Australian writer Stilgherrian told The Register that while it’d be easy to dismiss the claims, in spite of his eccentricities, Nelson “has the annoying habit of being right.”Simons Foundation article:

https://simonsfoundation.org/features/science-news/unheralded-mathematician-bridges-the-prime-gap/

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Just an update: Zhang’s paper is now available on the Ann. of Math. site: