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.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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