Mathematics
Perfect Numbers and Mersenne Primes
Quick fact
Every even perfect number is completely determined by a Mersenne prime, and the largest known perfect numbers are among the largest numbers in existence—each new one found corresponds to a brand-new Mersenne prime.
Why this is interesting
You know that 6 = 1 + 2 + 3, but did you know that this simple equality ties into the largest prime numbers ever discovered?
Read the full explanation
Understanding Perfect Numbers and Mersenne Primes
A perfect number is a positive integer that equals the sum of its proper divisors (all positive divisors except itself). The smallest is 6, because its proper divisors are 1, 2, and 3, and 1 + 2 + 3 = 6. The next is 28: 1 + 2 + 4 + 7 + 14 = 28. These have been known since ancient times. Now, the star connection is via Mersenne primes—primes of the form 2^p - 1, where p is itself a prime. For example, 2^2 - 1 = 3, 2^3 - 1 = 7, 2^5 - 1 = 31. Notice that when you compute 2^(p-1) (2^p - 1), you get 6, 28, and 496 respectively. That's the pattern that links the two concepts.
A deeper explanation
The Euclid-Euler theorem is the key: An even number is perfect if and only if it can be written as 2^(p-1) (2^p - 1), where 2^p - 1 is a Mersenne prime. Euclid proved the 'if' direction about 300 BC: if 2^p - 1 is prime, then the number N = 2^(p-1) (2^p - 1) is perfect. Two millennia later, Euler proved the converse: every even perfect number must be of this form. This theorem explains why the search for perfect numbers is exactly the search for Mersenne primes. Without it, perfect numbers would be random curiosities; with it, they are windows into the depths of prime numbers. Moreover, no odd perfect numbers have ever been found, and it remains an open question whether any exist, making this a living mystery that continues to drive research.