Editing Perfect number (section)

{{Short description|Number equal to the sum of its proper divisors}}
{{About||the 2012 film|Perfect Number (film){{!}}''Perfect Number'' (film)}}
[[File:Perfect number Cuisenaire rods 6 exact.svg|thumb|Illustration of the perfect number status of the number 6]]

In [[number theory]], a '''perfect number''' is a [[positive integer]] that is equal to the sum of its positive proper [[divisor]]s, that is, divisors excluding the number itself.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Perfect Number |url=https://mathworld.wolfram.com/PerfectNumber.html |access-date=2025-02-09 |website=mathworld.wolfram.com |language=en |quote=Perfect numbers are positive integers n such that  n=s(n), where s(n) is the restricted divisor function (i.e., the sum of proper divisors of n), ...}}</ref> 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 first four perfect numbers are [[6 (number)|6]], [[28 (number)|28]], [[496 (number)|496]] and [[8128 (number)|8128]].<ref>{{Cite web |title=A000396 - OEIS |url=https://oeis.org/A000396 |access-date=2024-03-21 |website=oeis.org}}</ref>

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, <math>\sigma_1(n)=2n</math> where <math>\sigma_1</math> is the [[sum-of-divisors function]].

This definition is ancient, appearing as early as [[Euclid's Elements|Euclid's ''Elements'']] (VII.22) where it is called {{lang|grc|τέλειος ἀριθμός}} (''perfect'', ''ideal'', or ''complete number''). [[Euclid]] also proved a formation rule (IX.36) whereby <math>q(q+1)/2</math> is an even perfect number whenever <math>q</math> is a prime [[of the form]] <math>2^p-1</math> for positive integer <math>p</math>—what is now called a [[Mersenne prime]]. Two millennia later, [[Leonhard Euler]] proved that all even perfect numbers are of this form.<ref name="The Euclid–Euler theorem">Caldwell, Chris, [https://primes.utm.edu/notes/proofs/EvenPerfect.html "A proof that all even perfect numbers are a power of two times a Mersenne prime"].</ref> 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.