Harald Helfgott has announced a proof of the odd Goldbach conjecture (also known as the ternary or weak Goldbach conjecture). This is big news. Like a good maths newshound, Christian Perfect promptly wrote this up for The Aperiodical as “All odd integers greater than 7 are the sum of three odd primes!”
Wait, though, there’s a problem. As Relinde Jurrius pointed out on Twitter, the formulation used in the paper abstract was not quite the same.
The ternary Goldbach conjecture, or three-primes problem, asserts that every odd integer $N$ greater than $5$ is the sum of three primes. The present paper proves this conjecture.
The version Christian used makes the assertion using odd primes, whereas the paper abstract only claims “the sum of three primes”. The latter version includes $7$ because $7$ can be written as the sum of three primes, but not odd ones ($7 = 3+2+2$). Certainly, you can see both statements of the weak Goldbach conjecture used (for example, here’s the $\gt 5$ version and here’s the $\gt 7$ version). Are they equivalent?
Here are some nice number facts and tricks you can try out on your friends. They will work without understanding how, but with a little investigation you should be able to figure out how each one works.
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+
“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
The Online Encyclopedia of Integer Sequences contains over 200,000 sequences. It contains classics, curios, thousands of derivatives entered purely for completeness’s sake, short sequences whose completion would be a huge mathematical achievement, and some entries which are just downright silly.
For a lark, David and I have decided to review some of the Encyclopedia’s sequences. We’ll be rating sequences on four axes: Novelty, Aesthetics, Explicability and Completeness.
Except for initial term, primes of form “n 3′s followed by 1″.
3, 31, 331, 3331, 33331, 333331, 3333331, 33333331, 333333333333333331, 3333333333333333333333333333333333333331, 33333333333333333333333333333333333333333333333331, ...
Update 14/05/2013: The seminar was successful: Zhang announced that his proof has already been refereed for the Annals, and everyone seems happy with it.
Hard Maths news now: there’s a rumour going round that Yitang (Tom) Zhang of the University of New Hampshire reckons he can prove that there are infinitely many different pairs of primes at most 70,000,000 apart.