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, because 28 has proper divisors 1, 2, 4, 7, 14, and 1 + 2 + 4 + 7 + 14 = 28. The first seven perfect numbers are 6, 28, 496, 8128, 33550336, 8589869056, and 137438691328 (sequence A000396 in the OEIS). 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, σ 1 ( n ) = 2 n {\displaystyle \sigma _{1}(n)=2n} where σ 1 {\displaystyle \sigma _{1}} is the sum-of-divisors function. This definition is ancient, appearing as early as Euclid's Elements (Book VII, Definition 22) where it is called τέλειος ἀριθμός (téleios arithmós; 'perfect', 'ideal', or 'complete number'). Euclid also proved a formation rule (Book IX, Proposition 36) whereby q ( q + 1 ) 2 {\textstyle {\frac {q(q+1)}{2}}} is an even perfect number whenever ⁠ q {\displaystyle q} ⁠ is a prime of the form 2 p − 1 {\displaystyle 2^{p}-1} for positive integer ⁠ p {\displaystyle p} ⁠—what is now called a Mersenne prime. Two millennia later, Leonhard Euler proved that all even perfect numbers are of this form. 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.