Perfect Number Checker Calculator
A perfect number is a positive whole number that equals the sum of its proper divisors, meaning every positive divisor except the number itself. The smallest example is 6, whose proper divisors 1, 2 and 3 add up to exactly 6. This checker takes any positive integer, finds and totals its proper divisors, and tells you whether the number is perfect, abundant (divisor sum too large) or deficient (divisor sum too small). It is a clean, self-contained piece of number theory with no empirical inputs.
Perfect number formula
Proper divisors of N = all d in 1..(N-1) where N mod d = 0
s(N) = sum of those proper divisors
Perfect if s(N) = N
Abundant if s(N) > N
Deficient if s(N) < N
The calculator tests each candidate divisor from 1 up to half of N (no divisor larger than N/2 can divide N evenly except N itself), adds the ones that divide evenly, and compares the total to N.
Facts about perfect numbers
- The first four perfect numbers are 6, 28, 496 and 8,128, all known since antiquity.
- Every even perfect number has the form 2^(p-1) times (2^p - 1) where 2^p - 1 is a Mersenne prime, a result proved by Euclid and Euler.
- No odd perfect number has ever been found despite extensive computer searches.
- Abundant numbers (divisor sum exceeds the number) are far more common than deficient or perfect numbers.
- The divisor sum here excludes the number itself, which is what distinguishes proper divisors from all divisors.
Perfect numbers: frequently asked questions
What is a perfect number?
A perfect number is a positive integer that equals the sum of its proper divisors (all positive divisors except the number itself). For example, 6 has proper divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is perfect.
What are the first few perfect numbers?
The first four perfect numbers are 6, 28, 496 and 8,128. The next is 33,550,336. Perfect numbers grow very quickly, and every known perfect number is even.
What does abundant or deficient mean?
If the sum of proper divisors exceeds the number, it is abundant (for example 12, whose divisors sum to 16). If the sum is less than the number, it is deficient (for example 8, whose divisors sum to 7). Equal means perfect.
Are there odd perfect numbers?
No odd perfect number has ever been found, and none exists below very large bounds verified by computation. Whether any odd perfect number exists is a famous unsolved problem in number theory.
How does the checker compute the divisor sum?
It tests each integer from 1 up to half the input. Every integer that divides the number with no remainder is a proper divisor, and the calculator adds those divisors together, then compares the total to the original number.
Official sources
- NIST Digital Library of Mathematical Functions: Functions of Number Theory.
- National Institute of Standards and Technology, Dictionary of Algorithms and Data Structures: Perfect number.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.