Perfect Number Checker

Perfect Numbers

A perfect number is a positive integer that equals the sum of all its proper divisors, meaning divisors smaller than the number itself.

Definition

n is perfect if: s(n) = n

Where s(n) = sum of all positive divisors of n except n itself.

The First Four Perfect Numbers

The next perfect number is 33,550,336.

Euclid-Euler Theorem

All known even perfect numbers take the form: 2^(p-1) × (2^p - 1), where (2^p - 1) is a Mersenne prime.

For example: 6 = 2¹ × 3, 28 = 2² × 7, 496 = 2⁴ × 31.

Open Questions

• Are there infinitely many perfect numbers? Unknown.
• Do any odd perfect numbers exist? None have ever been found, and none exist below 10^1500, but it remains unproven.

Related Concepts

• Deficient numbers: s(n) < n (most integers are deficient)
• Abundant numbers: s(n) > n (e.g., 12: s(12) = 16)
• Amicable numbers: s(a) = b and s(b) = a

History

The ancient Greeks knew the first four perfect numbers. Mersenne and Euler established their connection to Mersenne primes. Only 52 perfect numbers are known. The most recent came from the 52nd Mersenne prime, 2^136,279,841 − 1, found by Luke Durant for the GIMPS project in October 2024. The gaps between them are enormous: the search runs on distributed volunteer computing and each new one takes years.

One number is worth singling out. 1 is not perfect and not a special case of anything. Its only divisor is itself, so it has no proper divisors at all and s(1) = 0, which makes it deficient like most small integers.