Improved techniques for lower bounds for odd perfect numbers

Abstract:

If N is an odd perfect number, and ${q^k}||N$, q prime, k even, then it is almost immediate that $N > {q^{2k}}$. We prove here that, subject to certain conditions verifiable in polynomial time, in fact $N > {q^{5k/2}}$. Using this and related results, we are able to extend the computations in an earlier paper to show that $N > {10^{300}}$.