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All odd integers greater than 7 are the sum of three odd primes!

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+

3 Responses to “All odd integers greater than 7 are the sum of three odd primes!”

  1. Christian Perfect

    There’s been a bit of confusion over what exactly has been proved, due to the existence of two different formulations of “weak Goldbach conjecture”.

    The first, which I used for the title of this post and which came up the most often when I searched for “odd Goldbach conjecture”, goes as follows:

    Every odd integer greater than $7$ is the sum of three odd primes.

    The second, which Helfgott used in the abstract to “major arcs…”, and appears elsewhere in the literature, is:

    Every odd integer greater than $5$ is the sum of three primes.

    There are two differences: first of all, there’s no way of writing $7$ as the sum of three odd primes, but $7 = 3+2+2$, which is why the second formulation says “greater than $5$”.

    Secondly, the first formulation is a bit stronger than the second: it says you can always get away without using $2$ for bigger odd numbers. Clearly the “three primes” version of the statement follows from the “three odd primes” version, but it isn’t obvious that it works the other way round – might there be a big number which can be written as “odd prime + 4″, but not as the sum of three odd primes?

    Relinde Jurrius pointed out the existence of the different formulations to us on Twitter (I didn’t read the abstract very closely, it turns out!) and asked if they were equivalent. After much asking around, Harald Helfgott himself commented on Peter’s Google+ thread to reassure us:

    People, please don’t worry. I address this issue in pages 114-115 of the paper. I prove both versions of the conjecture, at the cost of about three lines of extra work.

    (And yes, one of the two versions implies the other, but, technically, not viceversa. I say “technically” because all partial results to date have worked equally well for both versions.)


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