Perfect number

In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number itself. For instance, 6 has proper divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is a perfect number. The next perfect number is 28, since 1 + 2 + 4 + 7 + 14 = 28.

The sum of proper divisors of a number is called its aliquot sum, so a perfect number is one that is equal to its aliquot sum. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors; in symbols, ${\displaystyle \sigma _{1}(n)=2n}$ where ${\displaystyle \sigma _{1}}$ is the sum-of-divisors function.

This definition is ancient, appearing as early as Euclid's Elements (VII.22) where it is called τέλειος ἀριθμός (perfect, ideal, or complete number). Euclid also proved a formation rule (IX.36) whereby ${\displaystyle q(q+1)/2}$ is an even perfect number whenever ${\displaystyle q}$ is a prime of the form ${\displaystyle 2^{p}-1}$ for positive integer ${\displaystyle p}$—what is now called a Mersenne prime. Two millennia later, Leonhard Euler proved that all even perfect numbers are of this form.^[1] This is known as the Euclid–Euler theorem.

It is not known whether there are any odd perfect numbers, nor whether infinitely many perfect numbers exist. The first few perfect numbers are 6, 28, 496 and 8128.^[2]