What are Perfect Numbers?
A perfect number is a positive integer that is equal to the sum of its proper divisors — that is, all of its positive divisors excluding itself. The smallest perfect number is 6, because its proper divisors are 1, 2, and 3, and 1 + 2 + 3 = 6. The next perfect number is 28, since 1 + 2 + 4 + 7 + 14 = 28.
Perfect numbers are exceedingly rare. Only a handful are known, and they grow astronomically large. The first four — 6, 28, 496, and 8128 — were known to the ancient Greeks. As of today, only 51 perfect numbers have been discovered, and each new discovery requires finding a new Mersenne prime.
Connection to Mersenne Primes
There is a deep connection between perfect numbers and Mersenne primes. Euclid proved that if 2p − 1 is prime (a Mersenne prime), then 2p−1(2p − 1) is a perfect number. For example, 22 − 1 = 3 is prime, giving the perfect number 21 × 3 = 6. Similarly, 23 − 1 = 7 is prime, giving 22 × 7 = 28.
Euler later proved the converse for even perfect numbers: every even perfect number must have this form. This is known as the Euclid–Euler theorem. Whether any odd perfect numbers exist remains one of the oldest unsolved problems in mathematics. No odd perfect number has ever been found, and it has been shown that if one exists, it must be greater than 101500.
Historical Significance
Perfect numbers have fascinated mathematicians for over 2,000 years. The ancient Greeks considered them to have mystical significance. Nicomachus of Gerasa (circa 100 AD) classified numbers as deficient, perfect, or abundant based on the sum of their proper divisors. He associated perfect numbers with virtue and beauty, deficient numbers with deprivation, and abundant numbers with excess.
In the medieval period, religious scholars attributed theological significance to perfect numbers. Saint Augustine noted that God created the world in 6 days, a perfect number, and that the lunar cycle is approximately 28 days, also a perfect number. These coincidences were seen as evidence of divine design.
Properties of Perfect Numbers
All known perfect numbers share remarkable properties. Every even perfect number ends in either 6 or 28 (when written in base 10). Every even perfect number is also a triangular number — it can be expressed as 1 + 2 + 3 + ... + n for some n. For example, 28 = 1 + 2 + 3 + 4 + 5 + 6 + 7. Every even perfect number greater than 6 is the sum of consecutive odd cubes: 28 = 1³ + 3³, 496 = 1³ + 3³ + 5³ + 7³.
The digital root of every even perfect number (except 6) is 1. The reciprocals of the divisors of a perfect number always sum to 2. For example, for 28: 1/1 + 1/2 + 1/4 + 1/7 + 1/14 + 1/28 = 2. Every even perfect number can be written in binary as a sequence of 1s followed by a sequence of 0s (e.g., 6 = 110, 28 = 11100, 496 = 111110000).
The Search for New Perfect Numbers
Finding new perfect numbers is equivalent to finding new Mersenne primes, which is the goal of the Great Internet Mersenne Prime Search (GIMPS). This distributed computing project has been running since 1996 and has discovered many of the largest known Mersenne primes. The largest known perfect number, as of recent discoveries, has tens of millions of digits.