Things move quickly in the math world. It was only the end of May that we heard of a stunning development regarding the long moribund twin primes conjecture. The problem is to prove that there are infinitely many pairs of prime numbers that differ by two, such as 3 and 5, or 17 and 19, or 101 and 103. The development was that a previously unknown mathematician named Yitang Zhang proved that there are infinitely many pairs of primes whose difference is…wait for it… no greater than seventy million. Seventy million is certainly a long way from two, but it is even farther away from infinity, and that’s why it was a significant advance. It was a sort of proof of concept for the twin primes conjecture.
That was a just a few months ago. But now a group of researchers led by Terence Tao, a mathematician at UCLA and a Fields Medalist, is announcing that they have refined Zhang’s methods to the point where instead of seventy million, the gap now stands at 4,680. Pretty good! But still a long way from two…
Actually, there has been almost daily progress on the problem ever since Zhang announced his results.
One of the frustrations of mathematical research is that if you are not a specialist in the relevant field, you are pretty much helpless to understand what is going on. Here’s an excerpt from the abstract of the paper that is in progress, reporting on these results:
The improvements used here include a numerical search for narrow admissible tuples; the use of an optimized sieve cutoff function (expressible in terms of a Bessel function) obtained recently by Farkas, Revesz, and Pintz; a relaxation of the Motohashi-Pintz-Zhang truncation of
the Goldston-Yildirim-Pintz sieve from smooth moduli to densely divisible moduli; an
efficient application of this truncation introduced recently by Pintz; a new approach
to estimation of Type III sums based on correlation bounds for hyper-Kloosterman
sums; more efficient use of the Weyl differencing method and further exploitation of
averaging in the moduli parameters; application of the q-van der Corput A-process of
Heath-Brown and Ringrose; and bounds on multidimensional exponential sums over
finite fields coming from Deligne’s work on the Weil conjectures.
Alas, I don’t know what most of that means. There are great and wonderful things happening in the world of mathematics, but even most mathematicians are not able to understand them. Oh well. Perhaps someone will eventually write a comprehensible trade book about all of this, just as happened eventually with the proof of Fermat’s Last Theorem.