Perfect Numbers In Math

525 bce studied perfect numbers for their mystical properties. The discovery of such numbers is lost in prehistory but it is known that the pythagoreans founded c. Perfect number a positive integer that is equal to the sum of its proper divisors.

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The first four perfect numbers were the only ones known to early greek mathematics and the mathematician nicomachus noted 8128 as early as around ad 100.

Perfect numbers in math. Euclid over two thousand years ago showed that all even perfect numbers can be represented by n 2p 1 2p 1 where p is a prime for which 2p 1 is a mersenne prime. A perfect number n is a positive integer which is equal to the sum of its factors excluding n itself. To find the proper factors of a number write down all numbers that divide the number with the exception of the number itself. A number is perfect if the sum of its proper factors is equal to the number.

The proper factors of 18 are 1 2 3 6 and 9 if the sum of the factors is equal to 18 then 18 is a perfect number. In about 300 bc euclid showed that if 2 p 1 is prime then 2 p 1 2 p 1 is perfect. The smallest perfect number is 6 which is the sum of 1 2 and 3. In modern language nicomachus states without proof that every perfect number is of the form where is prime.

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