hat M31 is prime. This established the record for another century and when Lucas showed that M127 (which is a 39 digit number) is prime that took the record as far as the age of the electronic computer.In 1952 the Mersenne numbers M521, M607, M1279, M2203 and M2281 were proved to be prime by Robinson using an early computer and the electronic age had begun. By 2001 a total of 39 Mersenne primes have been found. The largest is M13466917 which has 4053946 decimal digits. Euler's work had a great impact on number theory in general and on primes in particular. He extended Fermat's Little Theorem and introduced the Euler -function. As mentioned above he factorised the 5th Fermat Number 232 + 1, he found 60 pairs of the amicable numbers referred to above, and he stated (but was unable to prove) what became known as the Law of Quadratic Reciprocity.He was the first to realise that number theory could be studied using the tools of analysis and in so-doing founded the subject of Analytic Number Theory. He was able to show that not only is the so-called Harmonic series (1/n) divergent, but the series 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ... formed by summing the reciprocals of the prime numbers, is also divergent. The sum to n terms of the Harmonic series grows roughly like log(n), while the latter series diverges even more slowly like log[ log(n) ]. This means, for example, that summing the reciprocals of all the primes that have been listed, even by the most powerful computers, only gives a sum of about 4, but the series still diverges to . At first sight the primes seem to be distributed among the integers in rather a haphazard way. For example in the 100 numbers immediately before 10 000 000 there are 9 primes, while in the 100 numbers after there are only 2 primes. However, on a large scale, the way in which the primes are distributed is very regular. Legendre and Gauss both did extensive calculations of the density of primes. Gauss (who was a pr...
