Although Euclid’s argument does not tell exactly where to find the next prime number, the overall frequency of the primes is now quite well understood. For example, if we take any two numbers,a and b say, with no common factor and consider the sequence a,a+b,a+2b,a+3b.., it was shown by the German mathematician Johann Dirichlet(1805–59)that innitely many members of such a sequence are prime.(Of course, there is no hope if a and b do have a common factor, d say, as then every member in the list is also a multiple of d, and so is not prime.)When a=1 and b=2, we get the sequence of odd numbers which we know, by Euclid’s proof, contains innitely many prime numbers. Indeed, it can be shown through fairly simple adaptations of Euclid’s argument that other special cases such as the sequence of numbers of the forms 3+4n,5+6n, and 5+8n(as n runs through the successive values 1,2,3,...), each have innitely many primes. The general result of Dirichlet is, however,very difcult to prove.