This paper proves that an odd perfect number, which is a positive integer equal to the sum of its proper divisors, cannot exist if its prime factorization has certain properties. Specifically, if the odd perfect number N has distinct prime factors p and q_i's with p and the exponents congruent to 1 modulo 4, and an integer t divides 2β_i+1 for all i, then t^5 must divid...