Perfect Number Checker
Check whether an integer is a perfect number — equal to the sum of its proper divisors.
Lists all divisors, the sum, and known perfect numbers up to 10¹⁰.
Perfect Numbers
A perfect number is a positive integer that equals the sum of all its proper divisors — 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
| n | Proper Divisors | Sum |
|---|---|---|
| 6 | 1, 2, 3 | 6 ✓ |
| 28 | 1, 2, 4, 7, 14 | 28 ✓ |
| 496 | 1, 2, 4, 8, 16, 31, 62, 124, 248 | 496 ✓ |
| 8,128 | 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064 | 8,128 ✓ |
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. As of 2024, only 51 perfect numbers are known.