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Perfect number (view source)

Revision as of 20:09, November 22, 2008

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A '''perfect number''' is one for which the sum of its [[factor]]s (excluding the number itself) is equal to the given number.
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'''Examples''':
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The first perfect number is 6:
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* the factors of 6 are 1, 2, 3 and 6.
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* the sum of 1 + 2 + 3 = 6
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The second perfect number is 28:
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* the factors of 28 are 1, 2, 4, 7, 14 and 28
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* the sum of 1 + 2 + 4 + 7 + 14 = 28
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The next perfect number is 496:
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* the factors are : 1, 2, 4, 8, 16, 31, 62, 124, 248, 496
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* the sum of 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496
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It is relatively straightforward to show that any number '''''P''''' of the form  '''''P = 2<sup>n-1</sup> (2<sup>n</sup>-1)''''', for which ('''''2<sup>n</sup>-1''''') is prime, is a perfect number <ref> Proposition IX.36 of Euclid's Elements </ref>. Prime numbers of the form ('''''2<sup>n</sup>-1'''''), for which n is also prime, are called [http://primes.utm.edu/mersenne/ Mersenne Primes], after Marin Mersenne (1588-1648).  
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Note that ('''''2<sup>n</sup>-1''''') is not a prime for all prime n, for example ('''''2<sup>11</sup>-1''''') = 2047, which is divisible by 13. The first 12 Mersenne primes  are ('''''2<sup>n</sup>-1''''') with: n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107 and 127.
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Many of Euclid's successors assumed that all perfect numbers were of this form, [[Euler]] in  1849 provided the first proof that Euclid's formula gives all possible even perfect numbers.
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It is not known if any odd perfect numbers exist, although numbers up to '''''10<sup>300</sup>''''' have been checked without success.  
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== References ==
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<references/>
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[[Category:Mathematics]]
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