On the quantity m2−pkm^2-p^k where pkm2p^k m^2 is an odd perfect number -- Part II

Let $p^k m^2$ be an odd perfect number with special prime $p$. Extending previous work of the authors, we prove that the inequality $m < p^k$ follows from $m^2 - p^k = 2^r t$, where $r \geq 2$ and $\gcd(2,t)=1$, under the following hypotheses: (a) $m > t > 2^r$, or (b) $m > 2^r > t$. We also prove that the estimate $m^2 - p^k > 2m$ holds. We can also improve this unconditional estimate to $m^2 - p^k > {m^2}/3$.Comment: 11 pages, incorporated referee comments and suggestions, added Section 5.1, in press at NNTDM <https://nntdm.net/volume-28-2022/number-1/