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.^[1] 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, because 1 + 2 + 4 + 7 + 14 = 28.

The first seven perfect numbers are 6, 28, 496, 8128, 33550336, 8589869056, and 137438691328.^[2]

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 ${\textstyle {\frac {q(q+1)}{2}}}$ 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.^[3] 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.