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What are perfect numbers

  1. Jul 23, 2014 #1
    Definition/Summary

    A perfect number is a number which is the sum of its proper divisors (half the sum of its total divisors). Even perfect numbers are a Mersenne prime times a power of two; odd perfect numbers are not known to exist.

    Equations

    Sum-of-divisors function:
    [tex]\sigma(n)=\sum_{k|n}k[/tex]
    [tex]\sigma(p^aq^b)=\sigma(p^a)\sigma(q^b)\;\;(p,q\text{ relatively prime})[/tex]
    [tex]\sigma(p^a)=\frac{p^{a+1}-1}{p-1}[/tex]

    Definition of N perfect:
    [tex]2N=\sigma(N)[/tex]

    Form of an even perfect number:
    [tex]N=M_p(M_p+1)/2=2^{p-1}(2^p-1)[/tex]
    where M_p is a Mersenne prime.

    Extended explanation

    The first two perfect numbers are:

    6 = 1 + 2 + 3 = [itex]2^{2-1} (2^2-1)[/itex]

    28 = 1 + 2 + 4 + 7 + 14 = [itex]2^{3-1} (2^3-1)[/itex]

    The next two are:

    496 = [itex]2^{5-1} (2^5-1)[/itex]

    8128 = [itex]2^{7-1} (2^7-1)[/itex]

    * This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
     
  2. jcsd
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