Perfect Number Calculator
Determine whether any integer is perfect, abundant, or deficient based on its proper divisor sum. Use it to explore number theory or verify known perfect numbers like 6 and 28.
About this calculator
A perfect number equals the sum of its proper divisors (all positive divisors excluding the number itself). The classification works as follows: if the divisor sum σ(n) equals n, the number is Perfect; if σ(n) > n, it is Abundant; if σ(n) < n, it is Deficient. For example, 6 has proper divisors 1, 2, 3 and 1 + 2 + 3 = 6, making it perfect. The number 12 has divisors 1, 2, 3, 4, 6 summing to 16 > 12, so it is abundant. The algorithm checks every integer i from 1 to n−1 for divisibility, then sums the qualifying divisors. Known perfect numbers are rare — only 51 have been discovered as of 2024 — and all known ones are even, taking the form 2^(p−1) × (2^p − 1) when 2^p − 1 is prime.
How to use
Enter 28 in the 'Number to Check' field and enable 'Show Proper Divisors'. The calculator tests each integer from 1 to 27: divisors found are 1, 2, 4, 7, 14. Sum = 1 + 2 + 4 + 7 + 14 = 28. Since the divisor sum equals 28, the result is 'Perfect (divisor sum: 28)'. Now try 12: divisors are 1, 2, 3, 4, 6, summing to 16 > 12, so 12 is 'Abundant (divisor sum: 16)'. Try 9: divisors 1, 3 sum to 4 < 9, giving 'Deficient (divisor sum: 4)'.
Frequently asked questions
What makes a number perfect and how many perfect numbers are known?
A number is perfect when it exactly equals the sum of its proper divisors. The first four perfect numbers are 6, 28, 496, and 8128. As of 2024, only 51 perfect numbers have been discovered, all of them even. Euclid proved that 2^(p−1) × (2^p − 1) is perfect whenever 2^p − 1 is a Mersenne prime. It remains an unsolved problem in mathematics whether any odd perfect numbers exist.
What is the difference between abundant and deficient numbers?
An abundant number has a proper divisor sum greater than itself — for example, 12 has divisors 1, 2, 3, 4, 6 summing to 16. A deficient number has a divisor sum less than itself — for example, 9 has divisors 1 and 3 summing to only 4. Most integers are deficient; prime numbers are always deficient since their only proper divisor is 1. Abundant numbers, while less common than deficient ones, are infinite in number and include all multiples of 6 greater than 6.
Why are perfect numbers important in mathematics and number theory?
Perfect numbers connect number theory with Mersenne primes, one of the most actively researched areas of mathematics. Each new Mersenne prime discovered via the GIMPS distributed computing project simultaneously yields a new perfect number. Their study dates back to ancient Greece, where Euclid linked them to geometric series. Perfect numbers also appear in combinatorics and modular arithmetic, and the unsolved question of odd perfect numbers remains one of the oldest open problems in mathematics.