Perfect Numbers

What just happened?

A perfect number is a natural number that equals the sum of its proper divisors — that is, all its divisors except the number itself. For the number examined we summed its divisors and compared the result with the number itself. When the sum lands exactly on the number, we say the number is perfect.

The first two examples:

The proper divisors of 6 are 1, 2 and 3, and their sum:

The proper divisors of 28 are 1, 2, 4, 7 and 14, and their sum:

The remaining numbers split into deficient ($s(n) < n$, e.g. 8: $1+2+4 = 7 < 8$) and abundant ($s(n) > n$, e.g. 12: $1+2+3+4+6 = 16 > 12$). Perfect numbers lie exactly on the boundary between them.

Extremely rare and deeply tied to the primes

Perfect numbers have fascinated mathematicians since antiquity — the Pythagoreans ascribed mystical meaning to them, and Euclid studied them around 300 BC in the “Elements”. The first four are 6, 28, 496 and 8128; they are extremely rare — to this day we know only 51 of them.

Euclid's formula (and the Euclid-Euler theorem):

Euclid proved that if $2^p - 1$ is prime (a so-called Mersenne prime), then

is a perfect number. Check: for $p = 2$ we get $2^{1}(2^2-1) = 2\cdot 3 = 6$, and for $p = 3$ we get $2^{2}(2^3-1) = 4\cdot 7 = 28$. Euler later showed that all even perfect numbers have exactly this form. Whether odd perfect numbers exist is still unknown (an open problem in mathematics!).

Why this matters for your exam:

This is a model example of how one analyzes the divisors and divisibility of natural numbers — listing divisors, summing them and classifying numbers are typical school tools. Perfect numbers also show how deeply the theory of divisors connects with the Mersenne primes — unobvious structures hidden in ordinary natural numbers.

Remember: a number is perfect when the sum of its proper divisors (excluding itself) equals exactly that number — $s(n) = n$. The first are 6 and 28, and every even one is given by Euclid’s formula $2^{p-1}(2^p-1)$ with a Mersenne prime.