Arithmetic functions · Definition
Perfect number
Before you start you need to know
Perfect numbers have been studied since antiquity and are tied to Mersenne primes. Euclid and Euler characterized the even ones.
Definition
A perfect number is a positive integer \(n\) equal to the sum of its proper divisors, those smaller than \(n\); with the divisor-sum function \(\sigma\), the sum of all its divisors is twice \(n\).
Euclid proved that if \(2^p-1\) is prime, then \(2^{p-1}(2^p-1)\) is perfect, and Euler showed that every even perfect number has this form. Whether odd perfect numbers exist remains an open problem.
For \(p=2\), \(2^2-1=3\) is prime and \(2^{1}\cdot 3=6\) is perfect. For \(p=3\), \(2^3-1=7\) is prime and \(2^{2}\cdot 7=28\) is perfect too.
A common mistake is to add all divisors, including \(n\), and conclude that \(n\) is perfect when the sum equals \(2n\). To be perfect, the sum of the proper divisors must equal exactly \(n\).
Think about it
Why is 8 not perfect, even though it has four divisors?
Because its proper divisors 1, 2, and 4 sum to 7, not 8.
Exercises 10
- Which of these numbers is perfect?
- What is the second smallest perfect number?
- Which of these values of \(n\) satisfies the definition of a perfect number?
- How much do the proper divisors of 28 sum to?
- Which of these statements defines a perfect number \(n\)?
- Compute the sum of the proper divisors of 12 and state whether 12 is perfect.
- What is the third smallest perfect number?
- If \(2^p-1\) is prime with \(p=7\), which perfect number does Euclid's formula give?
- Which of the following statements is false?
- Let \(n=2^{10}(2^{11}-1)\). Is \(n\) perfect?
References
- David M. Burton, Elementary Number Theory, 7th edition, McGraw-Hill, 2011. cap. 11
- Ivan Niven, Herbert S. Zuckerman, Hugh L. Montgomery, An Introduction to the Theory of Numbers, 5th edition, Wiley, 1991. cap. 4