This Week in Number Theory

A visualization of the twin primes using an Ulam spiral. Created by Silveira Neto and shared under a Creative Commons-attribution-share alike license.

By now you’ve probably heard the announcements of two big results in number theory: Yitang Zhang of the University of New Hampshire proved that there are infinitely many pairs of primes whose differences are under 70 million, and Harald Helfgott of the École Normale Supériure in Paris posted his completion of the proof of the ternary, or weak, Goldbach conjecture to the arxiv. That conjecture states that every odd integer greater than 5 can be written as the sum of three primes.

Richard Elwes mentioned both results on his blog Simple City, as did Michael Lugo at God Plays Dice. The Aperiodical covered both announcements as well. The title of their Goldbach post, “All odd integers greater than 7 are the sum of three odd primes!” spurred a discussion on Twitter that then moved to Peter Rowlett’s Google Plus page about the difference between the two conjectures called “the weak Goldbach conjecture.” Number theorists, feel free to hop on over and enlighten us.

Twin primes

Zhang’s result on pairs of primes is a step towards the twin prime conjecture, which states that there are infinitely many primes that differ by only two. Peter Woit alerted the math-blog-o-sphere to the announcement in a post at Not Even Wrong on Sunday night: “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.” The updates and comments to Woit’s original post have a lot of additional information.

Perhaps the most pertinent addition is a link to Emily Riehl’s report from the talk at the n-Category Café group blog. Riehl, a postdoc at Harvard, is not a number theorist, but as she notes, “Whether we grow up to be category theorists or applied mathematicians, one thing that I suspect unites us all is that we were once enchanted by prime numbers.” Riehl’s post is a nice summary for those of us who aren’t experts in this field and just want an outline of the argument. The experts, or at least those experts who haven’t already read and/or refereed the paper, will have to wait until Zhang’s proof comes out in the Annals, as he hasn’t posted a preprint yet.

(In my procrastination surfing very important research in the n-Category Café archives, I found this post by John Baez from five years ago about the likelihood that various historical figures actually existed. “[D]id you know Homer didn’t write the Odyssey and Illiad? They were actually written by another guy with the same name!”)

David Roberts and Terry Tao both wrote public Google Plus posts about the twin prime conjecture. The announcement also inspired Silveira Neto to create a twin prime version of the Ulam spiral, which you can see at the top of this post. Patrick Honner, a high school math teacher in Brooklyn, uses the result to look at how big the gaps between primes can get. “And while being 70 million away may not seem close as far as prime numbers go, consider the following amazing fact: given any number N, we can find a string of N consecutive numbers that contains no primes at all!”

The announcement garnered some attention in broader science media, with Nature News, New Scientist, and io9 all covering the result.

Update, May 22: Erica Klarreich wrote an excellent article about twin primes for Simons Science News, and the New York Times covered the result as well. A preprint of Zhang’s article has been posted to the Annals website. The abstract is available for free, but the full article requires a subscription.

Goldbach

I wrote about Helfgott’s work on the ternary Goldbach conjecture on my blog, Roots of Unity. I was very pleased that Helfgott agreed to do an email interview with me, and I hope you’ll go read what he had to say about his work. My favorite quote from him was, “It was difficult to tell down the middle whether the plan would truly succeed. After all, if I had brought C down to 10¹⁰⁰, that would still have been larger than the number of subatomic particles in the universe multiplied by the number of seconds since the Big Bang-there would have been no hope of checking things that far!” A nice description of what it can feel like to be in the middle of the woods, hoping a lemma or estimate will get you all the way out.

Last year, Terry Tao proved that every odd integer is the sum of at most five primes, and his blog posts on his result and the heuristic limitations of the circle method will be of interest to those who want a much more thorough introduction to how people go about proving stuff about partitions of integers into primes. His Google Plus post about the latest news is also a helpful resource.

Truth is cool, a site I had never seen before, covered the Goldbach announcement, and Artem Kaznatcheev wrote about “the curse of computing” with respect to computer-assisted proofs like Helfgott’s, on the Theory, Evolution, and Games Group blog. Helfgott has a blog called The value of the variable, and I hope he’ll write something over there about his work.

ABC

This week’s optimistic number theory stories are a decided contrast to last week’s existential ruminations on the abc conjecture. Caroline Chen’s article The Paradox of the Proof, posted last Thursday, summarizes the frustration many feel with the opaqueness of Shinichi Mochizuki’s claimed proof. It is very well written and researched, although I do take issue with the idea that mathematicians are just sitting around waiting to see whether the conjecture holds up. I think most of us are busy working on other stuff!

Woit’s post about the twin prime conjecture (also linked above) touches on the abc issue, and Cathy O’Neil has two posts about it at mathbabe.org. Catarina Dutihl Novaes writes about what’s wrong with Mochizuki’s proof at the mathematical philosophy blog M-Phi. I am not well-versed in mathematical philosophy at all, so I found the ideas of adversariality and cooperation in proof interesting.

It seems that Mochizuki will be speaking about his work in Tokyo this June, and Go Yamashita, who has been trying to understand Mochizuki’s work, has written an FAQ on “Inter-universality.” Perhaps the community will  get some understandable answers about the abc conjecture sooner rather than later. And most importantly, Gil Kalai will find out whether Don Zagier owes him 1000 shekels.