perfect number

An positive integer n is called perfect if it is the sum of all positive divisorsMathworldPlanetmathPlanetmath of n less than n itself. It is not known if there are any odd perfect numbers, but all even perfect numbers have been classified according to the following lemma:

Lemma 1.

An even numberMathworldPlanetmath is perfect if and only if it equals 2k-1(2k-1) for some integer k>1 and 2k-1 is prime.


Let σ denote the sum of divisors function. Recall that this function is multiplicative.

Necessity: Let p=2k-1 be prime and n=2k-1p. We have that

σ(n) = σ(2k-1p)
= σ(2k-1)σ(p)
= (2k-1)(p+1)
= (2k-1)2k
= 2n,

which shows that n is perfect.

Sufficiency: Assume n is an even perfect number. Write n=2k-1m for some odd m and some k>1. Then we have gcd(2k-1,m)=1. Thus,


Since n is perfect, σ(n)=2n by definition. Therefore, σ(n)=2n=2km. Piecing together the two formulasMathworldPlanetmathPlanetmath for σ(n) yields


Thus, (2k-1)2km, which forces (2k-1)m. Write m=(2k-1)M. Note that 1M<m. From above, we have:

2km = (2k-1)σ(m)
2k(2k-1)M = (2k-1)σ(m)
2kM = σ(m)

Since mm by definition of divides ( and Mm by assumptionPlanetmathPlanetmath, we have


which forces σ(m)=m+M. Therefore, m has only two positive divisors, m and M. Hence, m must be prime, M=1, and m=(2k-1)M=2k-1, from which the result follows. ∎

The lemma can be used to produce examples of (even) perfect numbers:

  • If k=2, then 2k-1=22-1=3, which is prime. According to the lemma, 2k-1(2k-1)=22-13=6 is perfect. Indeed, 1+2+3=6.

  • If k=3, then 2k-1=23-1=7, which is prime. According to the lemma, 2k-1(2k-1)=23-17=28 is perfect. Indeed, 1+2+4+7+14=28.

  • If k=5, then 2k-1=25-1=31, which is prime. According to the lemma, 2k-1(2k-1)=25-131=496 is perfect. Indeed, 1+2+4+8+16+31+62+124+248=496.

Note that k=4 yields that 2k-1=24-1=15, which is not prime.

The sequenceMathworldPlanetmath of known perfect numbers appears in the OEIS as sequence njas/sequences/?q=A000396A000396.

Title perfect number
Canonical name PerfectNumber
Date of creation 2013-03-22 11:45:29
Last modified on 2013-03-22 11:45:29
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 24
Author Wkbj79 (1863)
Entry type Definition
Classification msc 11A05
Classification msc 20D99
Classification msc 20D06
Classification msc 18-00