A good new millenium for the primes

Prime numbers, the building blocks from which integers are made, are central to much of mathematics. Understanding their distribution is one of the most natural, and hence oldest, problems in mathematics. Once the ancient Greeks had determined that there are infinitely many then it was natural to ask how many there are up to any given point, perhaps a very large point, how many there are in certain special subsequences (for example, primes of the form “a square plus one”), and how to identify primes quickly. If one examines tables of primes then they appear to be “randomly distributed” though, as Bob Vaughan once put it, “we do not yet know what random means”. Answering these questions has thus proved to be difficult, each success requiring new, far reaching ideas and methods. During the last thirty years there had been few new results of this type but then, in the last decade, several surprises, some of which we will discuss here: 7 Identifying primes; 7 Primes values of some polynomials in two variables of degree > 2; 7 Arithmetic progressions of primes; 7 Short gaps between primes; 7 Links between the distribution of primes and randomly selected matrices.