Perfect Number Checker
Check whether a number is perfect, abundant, or deficient. Lists every proper divisor and the aliquot sum, the foundation of classical number theory.
🔢 What is a Perfect Number?
A perfect number is a positive integer that equals the sum of its proper divisors, the divisors that exclude the number itself. The classic example is 6: its proper divisors are 1, 2, and 3, and 1 + 2 + 3 = 6. The next perfect number is 28, whose divisors 1, 2, 4, 7, and 14 also sum to the number. Numbers that are not perfect fall into two other classes: abundant, where the divisor sum exceeds the number, and deficient, where it falls short.
These ideas come from classical number theory and have been studied since the ancient Greeks. Euclid described how to generate perfect numbers over two thousand years ago, and Euler later proved his formula captures every even perfect number. Beyond their history, divisor sums appear in cryptography, in the study of amicable and sociable numbers, and in problems about the distribution of integers. The classification of a number as perfect, abundant, or deficient is a natural first step in exploring its structure.
A common misconception is that perfect numbers are common. They are extremely rare: only 51 are known, and they grow astronomically large. Another misconception is that the number itself counts as one of its divisors when testing perfection. It does not; only proper divisors are summed. A third is that odd perfect numbers are known to exist. In fact, whether any odd perfect number exists remains an unsolved problem after more than 2000 years.
This checker takes any whole number, finds all of its proper divisors, adds them into the aliquot sum, and reports whether the number is perfect, abundant, or deficient. It also lists the divisors and shows the comparison, so you can see exactly why the number falls into its class.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — A Perfect Number
Is 28 a perfect number?
Example 2 — An Abundant Number
Is 12 abundant, perfect, or deficient?
Example 3 — A Deficient Number
Is 10 abundant, perfect, or deficient?
❓ Frequently Asked Questions
🔗 Related Calculators
What is a perfect number?
A perfect number is a positive integer that equals the sum of its proper divisors, the divisors excluding the number itself. The smallest is 6, whose proper divisors 1, 2, and 3 add up to 6. The next perfect numbers are 28, 496, and 8128. Perfect numbers have fascinated mathematicians since ancient Greece.
How do you check if a number is perfect?
Find all proper divisors of the number and add them up. If the sum equals the original number, it is perfect. For 28, the proper divisors are 1, 2, 4, 7, and 14, which sum to 28, so 28 is perfect. If the sum is less the number is deficient, and if more it is abundant.
What is an abundant number?
An abundant number is a positive integer whose proper divisors add up to more than the number itself. The smallest is 12, whose divisors 1, 2, 3, 4, and 6 sum to 16, which exceeds 12. Abundant numbers become more common as numbers grow larger, though they are still less frequent than deficient numbers.
What is a deficient number?
A deficient number is a positive integer whose proper divisors add up to less than the number itself. For example, 10 has proper divisors 1, 2, and 5, which sum to 8, less than 10. All prime numbers are deficient because their only proper divisor is 1. Most integers are deficient.
What is the aliquot sum?
The aliquot sum of a number is the sum of its proper divisors, meaning all positive divisors except the number itself. For 12 the aliquot sum is 1 + 2 + 3 + 4 + 6 = 16. Comparing the aliquot sum with the number determines whether it is perfect (equal), abundant (greater), or deficient (less).
What are the first perfect numbers?
The first four perfect numbers are 6, 28, 496, and 8128, all known to the ancient Greeks. The next are 33,550,336 and 8,589,869,056. Perfect numbers grow very quickly because each corresponds to a Mersenne prime, and only 51 are currently known.
Are there any odd perfect numbers?
No odd perfect number has ever been found, and none exists below at least 10^1500 according to computer searches. Whether any odd perfect number exists is one of the oldest unsolved problems in mathematics, dating back over 2000 years. All known perfect numbers are even.
How are perfect numbers related to Mersenne primes?
Every even perfect number has the form 2^(p−1) × (2^p − 1), where 2^p − 1 is a prime known as a Mersenne prime. Euclid proved this form always gives a perfect number, and Euler proved every even perfect number must take this form. So each Mersenne prime yields exactly one even perfect number.
What are proper divisors?
Proper divisors of a number are all its positive divisors except the number itself. For 28, the divisors are 1, 2, 4, 7, 14, and 28, but the proper divisors are just 1, 2, 4, 7, and 14. Proper divisors are used to define perfect, abundant, and deficient numbers through their sum.
Is 1 a perfect number?
No. The number 1 has no proper divisors, so its aliquot sum is 0, which is less than 1, making it deficient. The smallest perfect number is 6. By convention, perfect numbers are defined only for integers greater than 1 that equal the sum of their proper divisors.