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Thursday, August 21, 2014

Throwback Fact of the Week - Perfect Numbers - 8/21/14

Perfect Numbers

perfect number is a positive integer that is equal to the sum of its proper positive divisors. In other words, it is equal to the sum of its positive divisors excluding the number itself. The simplest example of this is the number 6; the proper divisors of 6 are 1, 2, and 3, 1+2+3=6, so 6 is a perfect number. 

Knowing that 6, which is a small integer, is a perfect number might lead one to believe that numbers of this form are common. This turns out to be false; they are much rarer. The next perfect numbers are 28, 496, and 8128. The next perfect number after 8128? 33,550,336. 

Perfect numbers are not new. Pythagoras and the Greeks knew about the first four perfect numbers (you can see how they might not have encountered or been able to find the factors of a number like 33,550,336). 

There have been a number of interesting results about perfect numbers. The most important result was initially discovered by Euclid and later expanded upon by Euler.

Euclid proved that if if 2p-1 is prime, then N = (2p-1)(2p-1) is an even perfect number. 

Prime numbers of the form 2p-1, where p is prime, are known as Mersenne primes, another bountiful world of study themselves. Two millennia later, Euler proved that (2p-1)(2p-1), where 2p-1 is a Mersenne prime, will produce all even perfect numbers (there is a 1-1 ratio).
It remains unknown if there are an infinite number of Mersenne primes (only 43 Mersenne primes are known so far). It also remains unknown if there are any odd perfect numbers.

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