Perfect Numbers

A perfect number is a positive integer that is equal to the sum of all its proper divisors. (A proper divisor of n is a number that exactly divides n, but it is not n itself.) For example, the smallest perfect number is 6, whose proper divisors are 1, 2 , and 3; indeed 6 = 1 + 2 + 3. The next perfect number is 28 (its proper divisors are 1, 2, 4, 7, 14, which sum to 28). The next two perfect numbers are 496 and 8128. Beyond that the values get really huge.

Only about 10 perfect numbers are known before 1900. And even now only 38 of them have been discovered. It is not yet known if there are infinitely many perfect numbers. All of the 38 perfect numbers discovered so far are even numbers. That is, since the subject was first studied by ancient Greeks over 2000 years ago, not a single odd number has been found to be a perfect number. Indeed, all odd numbers up to 10300 (at least) have been checked without finding any perfect number among them. Whether or not odd perfect numbers exist is another problem unsolved to this day.

As early as during Euclid's time, it was known that if a number of the form 2n - 1 is a prime, then the product 2n-1(2n - 1) is an even perfect number. Later, Euler proved that all even perfect numbers are of such form. Recall that a prime of the form 2n - 1 is called a Mersenne Prime (the first few are 3, 7, 31, and 127). Therefore, the fact that 2n - 1 is a prime is a necessary and sufficient condition that 2n-1(2n - 1) is an even perfect number. That is, each and every even perfect number corresponds to a distinct Mersenne prime, and vice versa. There are currently only 38 known Mersenne primes, so there are only 38 known perfect numbers. The largest perfect number corresponds to the Mersenne prime 26972593 - 1 (a number with over 2 million digits), the perfect number itself has 4197919 digits.

An interesting side note is about the binary representations of those numbers. Since the binary representation of 2k - 1 is just k 1's, and that multiplication of a binary number by 2k merely adds k 0's at the end (of the string of binary digits), we can see that:

The binary representation of every Mersenne prime is consisted of all 1's, e.g., 3 becomes 11, 7 becomes 111, 31 becomes 11111, etc.

The binary representation of every even perfect number is a string of 1's followed by a string of 0's, e.g., 6 becomes 110, 28 becomes 11100, and 496 becomes 111110000.

This connection between perfect numbers and Mersenne primes generated a great deal of interest in the study of the latter that has continued to this day and has long since outgrown its origin. Perhaps sometime in the future an article will be devoted to Mersenne numbers.

Next time: Values of a complex polynomials. Solution to the problem: show that every polynomial p(x) will take every complex value, i.e. for any complex number c, there exists a complex number a such that p(a) = c.