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.