Perfect numbers

A Perfect Number is a Whole Number where the sum of the Number’s proper divisors together with the Number 1 equals the Number itself.

The smallest Perfect Number is 6. Its proper divisors are 2 and 3. If we add them together with 1, we get 6:

$$ 1 + 2 + 3 = 6 $$

The next Perfect Number is 28:

$$ 1 + 2 + 4 + 7 + 14 = 28 $$

The next known Perfect Numbers are 496 and 8128.

It is still unknown whether any odd Perfect Numbers exist, and many believe they do not.

When we talk about a Number’s proper divisors, we mean all the Numbers it can be divided by where the result is a Whole Number.

Apart from 1 and the Number itself. These are called trivial divisors.

If we take the Number 12, it can be divided by 1, 2, 3, 4, 6 and 12.

Prime Numbers have no proper divisors, since they can only be divided by 1 and themselves.

If the sum of a Number’s proper divisors is smaller than the Number itself, the Number is called deficient.

If the sum of the divisors is greater than the Number itself, it is called an abundant Number.

For example, the Number 15 is deficient because:

$$ \large 1+3+5=9 \;\text{ and }\; 9\lt 15 $$

The Number 20 is abundant because:

$$ \large 1+2+4+5+10=22 \;\text{ and }\; 22\gt 20 $$

The most deficient Numbers are Prime Numbers, since they have no proper divisors and the sum is therefore always 1.

The first odd abundant Number is 945.

Perfect Numbers were already studied in antiquity.

Euclid showed that if $\, \large 2^p - 1$ is a Prime Number (a so-called Mersenne Prime), then $\, \large 2^{p-1}(2^p - 1)$ is a Perfect Number.

All known Perfect Numbers are of this form, and all are even.