Two recent major results in Number Theory

It’s a good week for mathematics. Two major breakthroughs in number theory were announced these past few days:

1. Yitang Zhang proved that there are infinitely many “twin primes”. The usual definition of twin primes is a pair primes that differ by 2 (e.g., 5 and 7, 41 and 43). The theorem proven by Zhang is weaker, where “twin primes” here means two primes that differ by less than 70 million. Despite the large constant, this is the first such result that (a) is not conditional on some other unproven conjecture (e.g. the Riemann hypothesis) being true, (b) involves a finite difference between primes.

In other words, the prime numbers do stick closely together indefinitely. Zhang has made the difficult step of taking down the infinite to the finite (though admittedly still a large number), so it’s just a matter of technical breakthrough to bring the 70 million down to 2, settling the centuries-old and notoriously difficult Twin Primes Conjecture.

Amazingly, Zhang is not a superstar or even well-known in this field, having a somewhat modest research career with only 2 articles in MathSciNet thus far. He is now in his 50s, holding a Lecturer post at University of New Hampshire. His triumph gives hope to those who believe that math is no more a young man’s game.

Despite the optimism, mathematicians are still holding their collective breath. Very few people have seen the paper (no preprints in arXiv!), so people have to wait for it to be published. There is a good reason for hope, though: two referees for Annals of Mathematics, the number one math research journal, have read the paper and found no error. Another reason for hope is given by Henryk Iwaniec, a top number theorist, who read the paper and claimed the proof is correct. It is reported that the proof uses well-known methods in a clever manner, rather than a totally new mathematical paradigm. Compare with the purported proof of ABC conjecture by Michozuki, which no one on Earth, other than the author, understood.


2. Harald Andres Helfgott finally settled the weak Goldbach Conjecture, which states that all odd integers greater than 5 can be written as a sum of three primes. The strong Goldbach Conjecture, or simply the Goldbach Conjecture, which states that all even integers greater than 2 can be written as a sum of two primes, remains unproven.

The Helfgott proof is a major breakthrough, following in the footsteps of Vinogradov, who showed in 1937 that the result is true for sufficiently large odd integers. Vinogradov himself did not give an explicit bound of “sufficiently large”; his student Borozdin later gave the humongous lower bound of 3 to the power of 3^15. Some heroic efforts took place in the last century to close this gap, with the objective of making the lower bound small enough so the small cases can be checked by computer. Two Chinese mathematicians got the bound 10^1346 in 2002, but it was still too large.

Helfgott uses the well-known Hardy-Littlewood circle method, and he succeeded due to his careful and tight bounds on the terms in the major and minor arcs of the circle. He managed to bring down Vinogradov “sufficiently large” lower bound to 10^30, and the small cases were checked satisfactorily by a computer.

Weak Goldbach Theorem now!!

These are the papers with the proof: &

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