Perfect Number Analyzer
Check whether a number is perfect, abundant, or deficient by comparing it to the sum of its proper divisors. Use it to explore number theory properties or scan a range of integers for perfect numbers.
About this calculator
A perfect number n equals the sum of its proper divisors (all divisors excluding n itself). Formally, σ(n) − n = n, where σ(n) is the sum of all divisors. If the sum of proper divisors s(n) > n the number is abundant; if s(n) < n it is deficient. All known perfect numbers are even and generated by the Euclid–Euler formula: n = 2^(p−1) × (2^p − 1), where 2^p − 1 is a Mersenne prime. The first four perfect numbers are 6, 28, 496, and 8128. Whether any odd perfect number exists remains one of the oldest unsolved problems in mathematics. The calculator computes s(n) by iterating over all integers from 1 to n−1 and summing those that divide n evenly.
How to use
Enter the number 28 in the 'Number to Analyze' field and select single-number mode. The calculator checks divisors: 1, 2, 4, 7, 14 all divide 28. Their sum = 1+2+4+7+14 = 28. Since s(28) = 28 = n, the number is classified as Perfect. Now try 12: divisors are 1, 2, 3, 4, 6, summing to 16 > 12, so 12 is Abundant. For a range search, enter 1 to 10000 to find that 6, 28, 496, and 8128 are the perfect numbers in that interval.
Frequently asked questions
What is the difference between a perfect, abundant, and deficient number?
These three categories classify every positive integer by comparing it to s(n), the sum of its proper divisors. A perfect number satisfies s(n) = n exactly — 6 and 28 are the smallest examples. An abundant number has s(n) > n, meaning its divisors 'overflow' the number itself; 12 is the smallest abundant number with s(12) = 16. A deficient number has s(n) < n; most primes and prime powers are deficient because they have few small divisors.
How many perfect numbers are known, and why are they so rare?
As of 2024, only 51 perfect numbers are known, all of them even. Their rarity stems from the Euclid–Euler theorem: every even perfect number requires a corresponding Mersenne prime 2^p − 1, and Mersenne primes themselves are extraordinarily rare. Finding a new Mersenne prime (a task requiring massive distributed computing via projects like GIMPS) automatically yields a new perfect number. No odd perfect number has ever been found, and it has been proven that any odd perfect number, if it exists, must exceed 10^1500.
Why is the number 6 considered the first perfect number?
The number 6 is perfect because its proper divisors — 1, 2, and 3 — sum to exactly 6. Ancient Greek mathematicians, including Euclid, recognised this property and considered 6 and 28 to have mystical significance. Euclid proved in Elements (Book IX) that numbers of the form 2^(p−1)(2^p − 1), when 2^p − 1 is prime, are always perfect. For p=2: 2^1 × 3 = 6, confirming 6 as the smallest perfect number. This formula remained the only known source of perfect numbers for over two thousand years.