Perfect Number Checker
Check if a number is perfect, abundant, or deficient and find all its proper divisors.
Known perfect numbers: 6, 28, 496, 8128, 33550336…
What Is the Perfect Number Checker?
The Perfect Number Checker determines whether a given integer is a perfect number — one that equals the sum of all its proper divisors (all positive divisors excluding itself). It also lists all proper divisors and shows why the number is or is not perfect.
Formula
How to Use
Enter any positive integer. The calculator finds all proper divisors (divisors less than the number itself), sums them, and compares the sum to the original number. If they are equal, the number is perfect. The tool also shows whether the number is deficient (sum < n) or abundant (sum > n).
Example Calculation
n=28: Divisors of 28: 1,2,4,7,14. Sum = 1+2+4+7+14 = 28 = n → PERFECT ✓. n=12: Divisors: 1,2,3,4,6. Sum=16 > 12 → ABUNDANT. n=8: Divisors: 1,2,4. Sum=7 < 8 → DEFICIENT.
Understanding Perfect Number Checker
Perfect numbers fascinated ancient Greek mathematicians, who believed they possessed special mystical properties. Euclid proved that 2^(p−1)(2^p−1) is perfect when 2^p−1 is prime (now called a Mersenne prime). Over 2000 years later, Euler proved that every even perfect number must have this form — yet whether odd perfect numbers exist remains one of mathematics' oldest open questions.
The study of perfect numbers led to the broader classification of integers by the ratio of divisor sums to the number itself: deficient (most numbers), perfect (extremely rare), and abundant (like 12, 18, 20). The related concept of amicable numbers (pairs where each equals the sum of the other's divisors) further extends this framework.
Perfect numbers are intimately connected to Mersenne primes — primes of the form 2^p−1. Finding new Mersenne primes (done through distributed computing projects like GIMPS) automatically yields new perfect numbers. As of 2024, the largest known Mersenne prime has over 41 million digits, making the corresponding perfect number even more astronomical.
Frequently Asked Questions
What are all known perfect numbers?
The first five perfect numbers are 6, 28, 496, 8128, and 33,550,336. As of 2024, only 51 perfect numbers are known — all are even. No odd perfect numbers have ever been found, and their existence is an unsolved problem.
What is the pattern in even perfect numbers?
Every even perfect number has the form 2^(p−1) × (2^p − 1), where 2^p − 1 is a Mersenne prime. For p=2: 2×3=6. For p=3: 4×7=28. For p=5: 16×31=496.
What is a deficient number?
A deficient number has the sum of its proper divisors less than itself. Most numbers are deficient: 4 (sum=1+2=3<4), 9 (sum=1+3=4<9), and all prime numbers (only proper divisor is 1).
What is an abundant number?
An abundant number has the sum of its proper divisors greater than itself. The smallest abundant number is 12 (1+2+3+4+6=16>12).
Is this calculator free?
Yes, completely free with no registration required.