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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: |
| − | Image:Hitler.jpg|"MOAR I SAY, MOAR!"
| + | * the factors of 6 are 1, 2, 3 and 6. |
| − | Image:Hitler.jpg|"MOAR I SAY, MOAR!"
| + | * the sum of 1 + 2 + 3 = 6 |
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| + | The second perfect number is 28: |
| − | Image:Hitler.jpg|"MOAR I SAY, MOAR!"
| + | * the factors of 28 are 1, 2, 4, 7, 14 and 28 |
| − | Image:Hitler.jpg|"MOAR I SAY, MOAR!"
| + | * the sum of 1 + 2 + 4 + 7 + 14 = 28 |
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| + | The next perfect number is 496: |
| − | Image:Hitler.jpg|"MOAR I SAY, MOAR!"
| + | * the factors are : 1, 2, 4, 8, 16, 31, 62, 124, 248, 496 |
| − | Image:Hitler.jpg|"MOAR I SAY, MOAR!"
| + | * 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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| − | </gallery> | |