Generate Perfect Numbers

What is a Perfect Number?

A perfect number is a positive integer that is equal to the sum of its positive proper divisors (all positive divisors excluding the number itself).

For example, 6 is a perfect number because its proper divisors are 1, 2, and 3, and:

The next perfect number is 28, because its proper divisors are 1, 2, 4, 7, and 14, and:

The Euclid-Euler Theorem

Euclid discovered that the first four perfect numbers are generated by the formula:

where $2^p - 1$ is a prime number (known as a Mersenne prime). Euler later proved that every even perfect number must be of this form.

Because Mersenne primes are extremely rare and grow extremely quickly, perfect numbers are also very sparse:

• The 3rd perfect number is 496 (for $p = 5$).
• The 4th perfect number is 8,128 (for $p = 7$).
• The 5th perfect number is 33,550,336 (for $p = 13$).
• The 8th perfect number is 2,305,843,008,139,952,128 (for $p = 31$).

Unsolved Mysteries

Despite centuries of study, some of the most fundamental questions about perfect numbers remain unsolved:

• Are there any odd perfect numbers? No odd perfect number has ever been found, and it has been proven that none exist below $10^{1500}$.
• Are there infinitely many perfect numbers? This is still an open question, directly tied to whether there are infinitely many Mersenne primes.