A Proof of the Odd Perfect Number Conjecture

The finiteness of the number of solutions to the rationality condition for the existence of odd perfect numbers is deduced from the prime decomposition of the product of repunits defined by the sum of the divisors, $\sigma(N)$. All of the solutions to this rationality condition satisfy the inequality ${{\sigma(N)}\over N}\ne 2$. This technique is then used to demonstrate that there are no odd perfect numbers $N=(4k+1)^{4m+1} \prod_{i=1}^\ell q_i^{2\alpha_i}$