Science Forums 2

[CraigD]

Originally Posted by Don Blazys

and is called a "perfect number" because:

.

Note that the proper factors of "multiply perfectly" as a "first root",
while the proper factors of "multiply perfectly" as a "second root".
If we continue in this manner, then we will find that the proper factors of
the next perfect number, , also "multiply perfectly", but as a "fourth root",
and that in fact, all perfect numbers "multiply perfectly" as some "Nth root".
(It's one of the "deeper" reasons why these numbers are called "perfect".)

Originally Posted by modest

What Don points out is not a property of perfect numbers and it's not "one of the "deeper" reasons why these numbers are called "perfect".". If fact, every number with an even number of factors will follow Don's little rule. I pulled 30 out of my hat, but consider its factors:
(1 • 30) = 30
(2 • 15) = 30
(3 • 10) = 30
(5 • 6) = 30
Each of these multiply to the number in question. So, the result of multiplying all the factors must be some multiple of the number being factored

I think Modest is correct.

Also, the perfect numbers are given the formula , where is an element of {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, ...}, OEIS sequence A000043, the same sequence that gives the Mersenne primes, . Therefore, the "N" in the "Nth root" of the product of the proper factors of every perfect number that Don describes will always be exactly 1 less its corresponding . For example, the product of the proper factors of the 10th perfect number,

is