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moved from Perfect Number
A '''perfect number''' is one for which the sum of its [[factor]]s (excluding the number itself) is equal to the given number.
'''Examples''':
The first perfect number is 6:
* the factors of 6 are 1, 2, 3 and 6.
* the sum of 1 + 2 + 3 = 6
The second perfect number is 28:
* the factors of 28 are 1, 2, 4, 7, 14 and 28
* the sum of 1 + 2 + 4 + 7 + 14 = 28
The next perfect number is 496:
* the factors are : 1, 2, 4, 8, 16, 31, 62, 124, 248, 496
* the sum of 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496
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>. Such numbers are called [[Mersenne]] primes, after Marin Mersenne (1588-1648).
The first 12 Mersenne primes are: n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107 and 127; each one of these will generate a perfect number when used in the formula above.
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.
It is not known if any odd perfect numbers exist, although numbers up to '''''10<sup>300</sup>''''' have been checked without success.
== References ==
<references/>
[[Category:Mathematics]]
'''Examples''':
The first perfect number is 6:
* the factors of 6 are 1, 2, 3 and 6.
* the sum of 1 + 2 + 3 = 6
The second perfect number is 28:
* the factors of 28 are 1, 2, 4, 7, 14 and 28
* the sum of 1 + 2 + 4 + 7 + 14 = 28
The next perfect number is 496:
* the factors are : 1, 2, 4, 8, 16, 31, 62, 124, 248, 496
* the sum of 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496
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>. Such numbers are called [[Mersenne]] primes, after Marin Mersenne (1588-1648).
The first 12 Mersenne primes are: n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107 and 127; each one of these will generate a perfect number when used in the formula above.
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.
It is not known if any odd perfect numbers exist, although numbers up to '''''10<sup>300</sup>''''' have been checked without success.
== References ==
<references/>
[[Category:Mathematics]]