Mathematics Alum Yitang Zhang proves Centuries-old Problem


Yitang Zhang

Yitang Zhang of the University of New Hampshire, who obtained his Ph.D. in 1991 from Purdue under the direction of Professor T.T. Moh, has proved a spectacular result in number theory. Announced in Nature on May 14, 2013, the result concerns prime numbers, specifically the gaps between consecutive primes.

The importance of prime numbers has been recognized since antiquity: any positive integer can be written as the product of prime numbers, essentially in a unique way, but prime numbers themselves cannot be written as the product of other (positive) integers. In this sense, prime numbers are the building blocks of all integers. Accordingly, research on prime numbers has been going on at least since the time of Euclid of Alexandria (around 300 B.C.); some of the deepest results of mathematics and some of the most difficult conjectures concern prime numbers—or are motivated by prime numbers. Prime numbers have found many applications outside mathematics as well. An early application from the time of the industrial revolution had to do with turbines: turbines with a prime number of blades are less prone to destructive resonances. More recently, factorizing very large numbers into primes has been used to assure secure transmission of data, something that is of great importance in cryptography and in credit card transactions.

The latter application depends on the existence of very large prime numbers. Here comes Euclid to the rescue. He found, and was able to prove rigorously, that there indeed exist arbitrarily large primes. In other words, there are infinitely many prime numbers. He also began to investigate how close prime numbers can be to each other. 2 and 3 are consecutive primes, but this is the only example of consecutive integers that are both prime (because one of two consecutive numbers must be even, and an even number is never prime, unless it is 2). Can prime numbers be two apart? Yes, for example 5 and 7 or 29 and 31 are two apart. Such pairs are called twin primes, and Euclid found many more pairs of twin primes. He was led to ask whether there are arbitrarily large twin primes, a problem that not only resisted all attempts to solve in the following 2,300 years, but one on which practically no progress has been made until very recently.

Zhang now solved what mathematicians would consider a slightly weakened version of the problem. He proved that there are infinitely many pairs of primes that are, if not two, then at most 70 million apart. This is a remarkable result, because in spite of many attempts by some of the greatest mathematicians of all times, very little has been known about gaps between consecutive primes. In 1850, the Russian mathematician P.L. Chebyshev in a landmark result verified a conjecture J. Bertrand had made 5 years earlier. According to Chebyshev's theorem, if p is a prime, the next prime will be less then 2p, so the gap between p and the next prime is less than p. Of course, as p gets larger and larger, the gap that this theorem can guarantee also gets larger and larger. Chebyshev's estimate applies to all primes, but the proof implied a rather better estimate on gaps that occur after certain primes that still form an infinite sequence. A slight improvement on the size of the gap followed from subsequent deep work of B. Riemann, J. Hadamard, and C. J. de la Vallée-Poussin from the late 19th century on the general statistics of prime numbers. However, these estimates on the size of the gap among consecutive primes still grew to infinity rather fast as the primes themselves grew. A new idea appeared in a 2005 paper by D. Goldston, J. Pintz, and C. Yildirim that lowered the estimate of the size of the gap below what follows from the statistics of prime numbers. This, however, still produced a gap that grew to infinity. Zhang saw how to improve on the Goldston-Pintz-Yildirim idea in a significant manner and was able to produce, for the first time, an estimate on the gap that did not grow to infinity: the gap is less than 70 million for infinitely many pairs of consecutive primes. His paper will appear in one of the very top mathematical journals, the Annals of Mathematics.

Simons Science News has another look at Zhang's Research.