# Last Sunday was a perfect day

Most geeky math types (like me) already know about pi day (March 14… 3/14, get it?).

Writing in The Times Online, Marcus du Sautoy suggests a new math holiday: June 28. He suggests calling this day the World Math Day (actually, he suggests World Maths Day).

Why? What is so mathematical about June 28?

June 28 can be expressed as 6/28 and 6 and 28 are the first two perfect numbers. A perfect number is an integer that is the sum of its factors (not including the number itself). For example, $6=1\cdot 2\cdot 3=1+2+3$ and $28=1\cdot 2\cdot 4\cdot 7 \cdot 14=1+2+4+7+14$. The next known perfect number is 496.

The timing for this pronouncement is especially fitting. A new perfect number was just discovered on April 12, 2009. Actually, what was found was the 47th Mersenne prime (as part of the Great Internet Prime Search). But I’m getting ahead of myself. Let me explain.

There is an intimate connection between perfect numbers and a special class of primes called the Mersenne primes. The ancient Greeks noticed that the first four perfect numbers (6, 28, 496, 8128) fit a pattern—they all had the form $2^{n-1}(2^n-1)$ for some $n$. Moreover, in each of these cases $2^n-1$ was prime (3, 7, 31, 127, respectively). In his Elements, Euclid proved that if $2^n-1$ is prime, then $2^{n-1}(2^n-1)$ is a perfect number. Then, two thousand years later Euler proved that every even perfect number must have this form. Primes of the form $2^n-1$ are called Mersenne primes.

Thus, each new Mersenne prime corresponds to a new perfect number.

By the way, we know of no odd perfect numbers, but they may exist. Thus there are exactly 47 known perfect numbers. It is not known whether there are finitely or infinitely many Mersenne primes.