Perfection in a number
It is of ten easy to find peculiar properties of small numbers that characterize themfor instance,3 is the only number that is the sum of all the previous numbers, while 2 is the only even prime(making it the oddest prime of all). The number 6 has a truly unique property in that it is both the sum and product of all of its smallerfactors:6=1+2+3=1×2×3.
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.
A Mersenne number mis one of the form 2p-1, where p is itselfa prime. If you take, by way of example, the first four primes,2,3,5,and 7, the first four Mersenne numbers are seen to be:3,7,31, and127, which the reader can quicklyv erify as prime. If p were not prime, suppose p=ab say, then m=2p-1 is certainly not prime either, as it can be verified that in these circumstances the number mhas 2a-1 as a factor. However, if p is prime then the corresponding Mersenne number is of ten a prime, or so it seems.