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The global picture of prime distribution is summed up by the observation of the leading 19th-century German mathematician and physicist Karl Friedrich Gauss(1777–1855)that p(n), the number of primes up to the number n, is approximately given by n/logn and that the approximation becomes more and more accurate as n increases. For example, ifwe take nto be one million,the ratio of n/logn suggests that, up to that stage, about one number in every 12.7 should be prime. Gauss’s observation, which in detail says something more precise, was not proved until 1896.The logarithm function referred to here is the so-called natural logarithm, which is not based on powers of 10, but rather on powers of a special number e, which is approximately equal to2.718. We shall hear more of this very famous number e in Chapter 6.