A perfect number is a positive integer, the sum of whose factors(except the number itself) add upto the number.
Example: 28 = 14 + 7 + 4 + 2 + 1
There are about 48 known perfect numbers at the time of writing this article.But none of them are odd. Computers have checked upto $10^{300}$ but could never find one. So I tried my luck disproving their existence.I had a little experience proving this (pefect squares have odd number of factors). But rather than disproving their existence I ended up finding constraints on their existence.
Let $x$ be a odd perfect number.
Then $x$ can be denoted as $x = p^a.q^b.r^c ..….$ where $p, q, r.....$ are prime factors.
Theorem 1: One and only one of the prime factors have an odd power i.e. only one of $a, b, c....$ is odd and all others are even.
Proof:
All the combination of factors of $x$ can be denoted by varying the powers of the prime factors.
The sum of all the factors of $x$ can then be written as
$s = (p^0 + p^1 + p^2 + ....p^a)(q^0 + ..... q^b)(r^0 + ...… r^c).....$
