Necessary Conditions For the Non-existence of Odd Perfect Numbers

We start with a result showing most odd cubes cannot be perfect numbers (see Theorem 1). Then we give a new proof of a special case of a result of Iannucci (see [IAN]) that shows that none of the even exponents in N ’s prime factorization can be congruent to 4 (mod 5) if 3|N (Theorem 2). We then extend that result by proving that certain sets of small primes, when taken to a large power, cannot divide an odd perfect number. This generates an upper bound on the number of small primes dividing certain odd perfect numbers (see Theorem 3-4 and Proposition 1).