Perfect Numbers
A perfect number equals the sum of its proper divisors (all positive divisors except itself). The smallest perfect numbers are 6 and 28.
Definition
A positive integer n is perfect if the sum of its proper divisors equals n. Proper divisors exclude n itself but include 1.
For 6, the proper divisors are 1, 2, and 3. Since 1 + 2 + 3 = 6, the number 6 is perfect.
Known Perfect Numbers
The first few perfect numbers are 6, 28, 496, and 8128. Each known even perfect number is tied to a Mersenne prime through the formula 2^(p−1) × (2^p − 1), where 2^p − 1 is prime.
No odd perfect numbers have been found, and it is unknown whether any exist. As of modern searches, all known perfect numbers are even.
Perfect vs. Abundant and Deficient
If the sum of proper divisors is less than n, n is deficient (e.g., 8: 1 + 2 + 4 = 7). If the sum exceeds n, n is abundant (e.g., 12: 1 + 2 + 3 + 4 + 6 = 16).
Perfect numbers sit exactly on the boundary where the divisor sum equals the number itself.
Examples
- 28 is perfect: 1 + 2 + 4 + 7 + 14 = 28.
- 6 is the smallest perfect number.
- 10 is not perfect: 1 + 2 + 5 = 8 ≠ 10.
FAQ
How many perfect numbers are known?
Over 50 even perfect numbers are known, each corresponding to a discovered Mersenne prime. New ones are found as larger Mersenne primes are identified.
Why are perfect numbers rare?
Perfect numbers require a very specific balance between a number and its divisors. Even perfect numbers depend on rare Mersenne primes, making them uncommon among integers.