Primes in arithmetic progression

Prime numbers have fascinated people since ancient times. Since the last century, their study has acquired importance also on account of the crucial role played by them in cryptography and other related areas. One of the problems about primes which has intrigued mathematicians is whether it is possible to have long strings of primes with the successive primes differing by a fixed number, namely arithmetic progressions of primes. This longstanding question has been settled recently, by Ben Green and Terence Tao, showing that in fact there are arithmetic progressions of any desired length, consisting only of prime numbers; thus for any k, there exist primes p1, p2, . . . , pk, such that the successive differences p2 – p1, . . . , pk – pk–1 are all equal. Once the existence is known for every k, it is automatic that there are infinitely many such strings for each k. Green is a postdoctoral fellow at the University of British Columbia, Vancouver, and Tao is a Clay Prize Fellow affiliated with the University of California at Los Angeles. Prior to the work of Green and Tao it was only known, thanks to the work of van der Corput in the late 1930s, that there are infinitely many strings of three primes each, in which the two successive differences coincide. Even for k = 4, such a statement remained open. The longest known arithmetic progressions of primes, known by computational methods, consist of 22 primes; two such strings are known, featuring primes with 14 and 15 digits (in decimal expansion), the successive differences being certain integers with 13 and 14 digits respectively. A 49-page manuscript of Green and Tao, currently under circulation, has been the subject of much interest in the mathematical community. It draws upon earlier deep work on arithmetic progressions in sets of integers with positive ‘density’, due to Szemeredi, Furstenberg, Gowers and others, on the one hand, and some recent results of Goldston and Yildirim on prime members, on the other hand.