Upper bounds on the second largest prime factor of an odd perfect number

摘要

Acquaah and Konyagin showed that if N is an odd perfect number with prime factorization N = p(1)(a1) p(2)(a2) ... p(k)(ak), where p(1) p(2) ... p(k), then one must have pk 3(1/3) N-1/3. Using methods similar to theirs, we show that p(k-1) (2N)(1/5) and that p(k-1)p(k) 6(1/4) N-1/2. We also show that if p(k) and p(k-1) are close to each other, then these bounds can be further strengthened.