What this calculator does
A perfect number equals the sum of its own proper divisors. 6 is the smallest, since 1 plus 2 plus 3 is exactly 6, and 28 is the next, being 1 plus 2 plus 4 plus 7 plus 14.
They are extraordinarily rare. Only 51 are known, all even, and whether any odd perfect number exists is one of the oldest unsolved problems in mathematics. Almost every number is abundant or deficient instead.
The formula
All divisors below the number itself are found and summed. That sum is compared against the number: equal is perfect, greater is abundant, less is deficient.
| Term | Meaning |
|---|---|
| Perfect | Equal to the sum of its proper divisors. |
| Abundant | Less than the sum of its proper divisors, such as 12. |
| Deficient | Greater than that sum, which most numbers are. |
The inputs explained
| Field | What to enter |
|---|---|
| Whole number | The whole number to classify. Must be at least 1. |
When to use it
Exploring number theory
Perfect numbers are one of the oldest topics in the subject.
Finding divisors quickly
The full list of proper divisors is shown alongside the classification.
Checking a claim
Testing whether a number really is perfect takes one calculation.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How do numbers classify?
Each compared against its own divisor sum.
Questions
How many perfect numbers are known?
Fifty-one as of recent counts, and each corresponds to a Mersenne prime. New ones are found only when new Mersenne primes are discovered, which now happens very rarely and requires enormous computation.
Are there odd perfect numbers?
Nobody knows. It is one of the oldest open problems in mathematics. If one exists it has been shown to be larger than 10 to the power 1500 and to satisfy many other restrictive conditions, but no proof either way exists.
What is the connection to Mersenne primes?
Euclid proved that if 2 to the p minus 1 is prime, then that value times 2 to the p minus 1 is perfect. Euler later proved every even perfect number has exactly that form, which ties the two problems together completely.
Are most numbers abundant or deficient?
Deficient, by a clear margin. Abundant numbers have a natural density of roughly a quarter, perfect numbers are vanishingly rare, and everything else is deficient.
For pairs that sum to each other, see the amicable numbers calculator. For divisibility rules, see the divisibility test calculator.