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
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 e3100, and Helfgott’s proof reduces that bound to 1030 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: 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: Nice piece about this in Slate from Jordan Ellenberg.
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