Perfect Numbers

A perfect number, in mathematics, is a whole number that is equal to the sum of its divisors including 1 but excluding the number itself. Thus, for example, 6 is a perfect number because 1 + 2 + 3 = 6. Likewise 28 is a perfect number because 1 + 2 + 3 + 4 + 7 + 14 = 28. The terminology of perfect goes back to the ancient Greeks of Euclid of Alexandra’s (c. 325–c. 265 BC) time who would personify numbers. Thus, if the sum of the divisors was less than the number, the number was called deficient. If the sum was greater than the number it was called abundant. The ancient Greeks had calculated the first four perfect numbers: 6, 28, 496, and 8, and 128.

Swiss mathematician Leonhard Euler (1707– 1783)—who was born in Switzerland but worked in Germany and Russia—proved that every even perfect number is of the form 2^p–1 (2p –1) where p is a prime and 2^p–1 is also a prime. He called a Mersenne prime in honor of French mathematician and philosopher Marin Mersenne (1588–1648), a Franciscan friar who often served as an intermediary in the correspondence between the most prominent mathematicians of his time. For example, if p = 2, a prime, then 2^p–1 = 2² –1 = 4 – 1 = 3 is also a prime and 2^p–1 (2^p –1) = 2 × 3 = 6 that, as has been seen is indeed a perfect number. If p = 3, the next prime, then 2^p –1 = 2³ –1 = 8 – 1 = 7 is again a prime. This gives 2² × 7 = 4 × 7 = 28 which, as been seen, is also a perfect number. The first case where p is a prime but 2^p – 1 is not occurs when p = 11. Here we have 2¹¹ –1 = 2048 – 1 = 23 × 89.

The search for even perfect numbers, then, is the same as the search for Mersenne primes. As of October 2006, 44 are known corresponding to p = 2, 3, 7, 13,..., 756,839,..., 2^32,582,656 × (2^32,582,657 – 1) where the last one, as well as some of the others, were found by computers. The largest of these, 2^32,582,656 × (2^32,582,657 – 1) yields a perfect number of 19,616,714 digits.

At the present time, no odd perfect numbers are known. It is suspected that none exist and this hypothesis has been tested by computers up to their limits but, of course, this does not constitute a proof that none exist.

Roy Dubish