Perfect Numbers

A perfect number N is defined as any positive integer where the sum of its divisors except the number itself equals the number. The first few of these, which were known and respected by the ancient Greeks, are 6, 28, 496, and 8128.

Perfect number formula

Over two thousand years ago, Euclid showed that all even perfect numbers can be represented by:

$N=2^{p-1}(2^p-1)$ where p is a prime for which $2^p-1$ is a Mersenne prime.

That is to say, we have an even perfect number of the form N whenever the Mersenne Number $2^p-1$ is a prime number. Mersenne was obviously familiar with Euclid's book when coming up with his primes.

Perfect number table

How many perfect numbers are there?

There are 51 known perfect numbers. There are only two perfect numbers from 1 through 100, which are 6 and 28. The most recently discovered perfect number was discovered in 2018 and it has 49,724,095 digits. So far, all known perfect numbers are even. It is not known if there are any odd perfect numbers, but none have been found so far.

Solved examples of perfect numbers

Find all the perfect numbers from 1 to 500.

Example 1

We know that every perfect number can be expressed as $N=2^{p-1}(2^p-1)$ where p is a prime

number. We can use this formula to find all the perfect numbers from 1 to 500.

For $n=2,2^{2-1}(2^2-1)=2\left(4-1\right)=6$

For $n=3,2^{3-1}(2^3-1)=2^2\left(8-1\right)=4\times 7=28$

For $n=5,2^{5-1}(2^5-1)=2^2\left(32-1\right)=16\times 31=496$

And as we know from the table above, the perfect numbers from 1 to 500 are 6, 28, and 496.

Example 2

Check whether the following numbers are perfect numbers or not.

a. 282

The factors of 282 are 1, 2, 3, 6, 47, 94, 141, and 282

The proper divisors of 282 are 1, 2, 3, 6, 47, 94, and 141

$1+2+3+6+47+94+141=294$

Since 294 (does not equal) 282, then 282 is not a perfect number.

b. 8128

The factors of 8128 are 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064, and 8128