Perfect Number Calculator
Check if a number is perfect, deficient or abundant.
Results update as you type.
How it works
Whether a number is perfect, deficient, or abundant depends on how its divisors add up, and this calculator determines it. You enter a whole number, and it returns the sum of its proper divisors and its classification. Proper divisors are all the positive numbers that divide it evenly, excluding the number itself. A perfect number equals the sum of its proper divisors: the smallest is 6, whose divisors 1, 2, and 3 sum to 6, followed by 28 (1 + 2 + 4 + 7 + 14). If the divisor sum is less than the number, it is deficient (most numbers, like 8 or any prime), and if greater, it is abundant (like 12, whose divisors sum to 16). Perfect numbers are rare and beautiful objects in number theory, known since antiquity; every known perfect number is even and connected to Mersenne primes through a formula proved by Euclid and Euler, and it is still unknown whether any odd perfect numbers exist. The calculator finds the divisor sum efficiently by checking divisors only up to the square root, so it works quickly even for large numbers. This is useful for math students exploring number theory, for recreational mathematics, and for checking these classifications. The number must be a positive integer. Enter a number to find its divisor sum and whether it is perfect, deficient, or abundant.
Compare the sum of proper divisors to the number itself
A number's proper divisors are all its positive divisors except itself. If their sum equals the number, it is perfect; if the sum is less, deficient; if more, abundant. The calculator sums the proper divisors efficiently and classifies the number.
Source:Number theory
We review all our tools from time to time to ensure accuracy.
Worked example
How to use this calculator
- 1
Enter your values
Enter number.
- 2
Adjust the options
Fine-tune any options to match your situation.
- 3
Read your result
Classification updates instantly as you type — no button to press.
Frequently asked questions
What is a perfect number?
A perfect number is a positive integer that equals the sum of its proper divisors, all its divisors except itself. The smallest is 6 (1 + 2 + 3), followed by 28, 496, and 8,128. Perfect numbers are rare and have fascinated mathematicians since ancient Greece for their elegant balance.
What are deficient and abundant numbers?
A deficient number has proper divisors that sum to less than the number itself, which is true of most numbers, including all primes. An abundant number has proper divisors summing to more than the number, like 12, whose divisors 1, 2, 3, 4, and 6 sum to 16. Perfect numbers sit exactly between these two categories.
How are proper divisors summed?
Proper divisors are the positive divisors of a number excluding the number itself. The calculator finds them efficiently by testing potential divisors only up to the square root, adding both the divisor and its paired quotient. This makes checking even large numbers fast. It then compares the sum to the number to classify it.
Are there odd perfect numbers?
No odd perfect number has ever been found, and whether any exist is one of the oldest unsolved problems in mathematics. All known perfect numbers are even and correspond to Mersenne primes through a formula proved by Euclid and Euler. Extensive searches have ruled out odd perfect numbers below very large bounds, but none is proven impossible.
How are perfect numbers related to Mersenne primes?
Euclid showed, and Euler proved the converse, that every even perfect number has the form two to the power (p minus 1) times (two to the power p minus one), where the second factor is a Mersenne prime. So each Mersenne prime yields exactly one even perfect number, linking these two famous topics in number theory.