Mersenne composites and cyclotomic primes

Extract

One of the long-standing problems of number theory, appealing to professional and recreational mathematicians alike, is the existence of Mersenne primes. These puzzling primes, for example 7, 31, 127 and 8191, are of the form 2^P - 1, where p is itself a prime. The problem of their existence originated some 2400 years ago with the early Greek mathematicians' quest for the so-called perfect numbers, those like 6 and 28 which are the sum of their proper divisors. The connection was given in Euclid's Elements in 300 BC: if 2^P - 1 is prime, then 2^P-1(2^P - 1) is a perfect number. Much later, Euler proved that all the even perfect numbers correspond to Mersenne primes. So the interest for many years has been in finding Mersenne primes. Only 39 are known, including several monsters discovered in recent years using thousands of PCs coordinated via the internet (see [1] for information on the project). Many of us would like to know if there is any way of predicting which exponents yield Mersenne primes and whether there is an infinite number of them.