Perfect Number Calculator
This perfect number calculator tells you whether any whole number is perfect, deficient or abundant, based on the classic definition used in number theory: a perfect number is one whose proper divisors, every divisor except the number itself, add up to exactly that number. Enter a whole number between 2 and 1,000,000,000 and the calculator finds every proper divisor, sums them, and returns the total alongside a classification. If the sum of proper divisors equals your number it is perfect, like 6 (1 + 2 + 3) or 28 (1 + 2 + 4 + 7 + 14); if the sum is higher the number is abundant, and if lower it is deficient. The result card shows both the sum of proper divisors and the classification at a glance, so you can quickly check a single number or compare several. Perfect numbers are rare: only a handful are known, all of them even, and they are closely linked to Mersenne primes through a result proved by Euclid and later completed by Euler. Whether an odd perfect number exists remains an unsolved problem in mathematics. Use this tool to explore number theory, check homework, verify a claim, or simply satisfy curiosity about a number that catches your eye. The worked example below shows 28 confirmed as perfect and 12 shown to be abundant.
The formula
Add up all the proper divisors of n (every divisor except n itself). If the sum equals n the number is perfect; if it is more, n is abundant; if less, n is deficient.
Worked example
The proper divisors of 28 are 1, 2, 4, 7 and 14, which add to 28, so 28 is perfect. The number 12 has proper divisors adding to 16, so it is abundant. Enter 28 to confirm.
Frequently asked questions
What is a perfect number?
A number equal to the sum of its proper divisors, like 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14.
What are deficient and abundant numbers?
Deficient means the proper divisors add up to less than the number, abundant means they add up to more.
Are there odd perfect numbers?
None has ever been found, and whether any exist is a famous open question. All known perfect numbers are even.
Who this calculator is for
This calculator is for students, programmers and anyone exploring number theory.
What this calculator assumes
- You enter a whole number within the stated range.
- The exact integer algorithm is used.
- Very large results are shown grouped for readability.
Formula and sources
Related calculators
- Prime Factorization Calculator: break a number into prime factors.
- LCM Calculator: lowest common multiple.
- GCF Calculator: greatest common factor.