How to Calculate Perfect Numbers

Six is the smallest perfect number.

A perfect number is any number that is equal to the sum of its positive divisors. For example, six can be evenly divided by one, two and three and it is also the sum of one, two and three, so six is a perfect number. Perfect numbers can be calculated with Mersenne primes.

Function

Mersenne primes are prime numbers that are formed by raising 2 to a power k and subtracting 1; in equation form that would be m = (2^k) -1. Multiplying the Mersenne prime (2^k) -1 and 2^(k - 1), which may or may not be a prime, gives a perfect number.

Considerations

There are no known odd perfect numbers. Every number up to 300 digits long has been checked, but so far no one has proven that an odd perfect number cannot exist, so the possibility must remain open.

Fun Fact

Social numbers are very similar to perfect numbers, but they must come as a pair. A social number is equal to the sum of its partner's divisors. For example, 284 is equal to the sum of the divisors of 220, and 220 is equal to the sum of the divisors of 284. This makes 220 and 284 social numbers.

References

• The Prime Glossary: Perfect Number
• "A Study of Perfect Numbers and Related Topics, With Special Emphasis on the Search for an Odd Perfect Number"; J.W. Gaberdiel

Resources

• "A Friendly Introduction to Number Theory"; Joseph H. Silverman