The Pythagoreans called a number like 6perfect, meaning that the number is the sum of its proper factors, as we shall call them,which are the divisors strictly smaller than the number itself. This kind of perfection is indeed very rare. The first five perfect numbers are 6,28,496,8128, and 33,550,336. Alot is known about the even perfect numbers but, to this day, no one has been able to answer the basic question of the Ancients as to whether there are infinitely many of these special numbers. What is more,no one has found an odd one, nor proved that there are none.Any odd perfect number mustbe extremelylarge and there is a long list of special properties that such a number must possess in consequence of its odd perfection. However, all these restrictions have not as yet legislated such a number out of existenceconceivably, these special properties serve to direct our search for the elusive first odd perfect number, which may yet be awaiting discovery.The even perfects were known to Euclid to have a tight connection with a very special sequence of primes, known to us as the Mersenneprimes named after Marin Mersenne(1588–1648),a 17th-century French monk.