A famous theorem of EUCLID [VI.2] asserts that there are infinitely many primes. But what if one wants more information about these primes? For instance, are there infinitely many primes of the form 4n - 1? A fairly straightforward modification of Euclid’s argument shows that there are, and a slightly more difficult modification proves that there are infinitely many of the form 4n + 1 as well. However, modifications of Euclid’s argument are not enough to prove the general result in this direction, which is that if a and m are coprime (that is, have highest common factor 1), then there are infinitely many primes of the form mn + a. This was proved by DIRICHLET [VI.36] using what are now called Dirichlet L-FUNCTIONS [III.47], which are closely related to the RIEMANN ZETA FUNCTION [IV.2 §3]. The condition that m and a have highest common factor 1 is clearly necessary, since any common factor of m and a will be a factor of mn + a. Dirichlet’s theorem is discussed further in ANALYTIC NUMBER THEORY [IV.2 §4].