Perfect Number Checker
Our free number theory calculator solves perfect number problems. Get worked examples, visual aids, and downloadable results.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
Perfect: sigma(n) - n = n, i.e., sum of proper divisors = n
A number n is perfect when the sum of all its proper divisors (divisors less than n) equals n itself. The divisor function sigma(n) sums all divisors including n. For even perfect numbers: n = 2^(p-1) * (2^p - 1) where 2^p - 1 is a Mersenne prime.
Worked Examples
Example 1: Checking if 28 is a Perfect Number
Problem:Find all proper divisors of 28 and determine if their sum equals 28.
Solution:Proper divisors of 28: 1, 2, 4, 7, 14\nSum = 1 + 2 + 4 + 7 + 14 = 28\n28 = 28, so 28 IS a perfect number.\nMersenne connection: 28 = 2^2 * (2^3 - 1) = 4 * 7\nMersenne prime: 7 = 2^3 - 1 with exponent p = 3
Result:28 is PERFECT | sigma(28) = 56 | Abundancy index = 2.0000
Example 2: Classifying the Number 12
Problem:Determine whether 12 is deficient, perfect, or abundant.
Solution:Proper divisors of 12: 1, 2, 3, 4, 6\nSum = 1 + 2 + 3 + 4 + 6 = 16\n16 > 12, so 12 is ABUNDANT.\nAbundance = 16 - 12 = 4\nAbundancy index = sigma(12)/12 = 28/12 = 2.333
Result:12 is ABUNDANT | Sum of divisors: 16 | Abundance: +4
Frequently Asked Questions
What is a perfect number and how is it defined?
A perfect number is a positive integer that equals the sum of its proper divisors, which are all divisors excluding the number itself. The first four perfect numbers are 6 (1+2+3), 28 (1+2+4+7+14), 496, and 8128. These numbers have fascinated mathematicians since antiquity. Euclid proved around 300 BCE that if 2^p - 1 is prime (a Mersenne prime), then 2^(p-1) * (2^p - 1) is a perfect number. Euler later proved that all even perfect numbers have this form. Whether any odd perfect numbers exist remains one of the oldest unsolved problems in mathematics, with no examples found despite extensive computer searches up to 10^2200.
How are perfect numbers connected to Mersenne primes?
The Euclid-Euler theorem establishes a beautiful one-to-one correspondence between even perfect numbers and Mersenne primes. Euclid showed that whenever 2^p - 1 is prime, the number 2^(p-1) * (2^p - 1) is perfect. Two thousand years later, Euler proved the converse: every even perfect number has exactly this form. So 6 = 2^1 * 3 corresponds to the Mersenne prime 3 = 2^2 - 1, and 28 = 2^2 * 7 corresponds to the Mersenne prime 7 = 2^3 - 1. Since there are currently 51 known Mersenne primes, there are exactly 51 known even perfect numbers. Finding a new Mersenne prime automatically yields a new perfect number.
Do odd perfect numbers exist?
The existence of odd perfect numbers is one of the most famous unsolved problems in number theory, open for over 2,000 years. No odd perfect number has ever been found, and most mathematicians believe none exist. However, a proof of their nonexistence has eluded researchers. If one exists, it must satisfy stringent conditions: it must be greater than 10^2200, have at least 101 prime factors (counting multiplicity), have a special prime factor of the form p^a where p is congruent to 1 modulo 4, and cannot be divisible by 105. Despite these constraints, a complete proof remains elusive. The problem illustrates how seemingly simple definitions can lead to extraordinarily difficult mathematical questions.
What are multiply perfect numbers?
A multiply perfect number (or multiperfect number) is a positive integer n whose divisor sum sigma(n) is a multiple of n. Ordinary perfect numbers are 2-perfect since sigma(n) = 2n. The smallest 3-perfect number is 120, with sigma(120) = 360 = 3 * 120. The smallest 4-perfect number is 30,240, and the smallest 5-perfect number is 14,182,439,040. As the multiplicity k increases, the numbers become extraordinarily large and sparse. There are only about 6 known 11-perfect numbers. Finding and studying multiply perfect numbers requires sophisticated factoring algorithms and extensive computation. They connect to questions about the structure of the divisor function in analytic number theory.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy