A Proof That There Are No Odd Perfect Numbers
Recall that we have from [Dris, Dagal (2021) - discussion of the proof of Theorem 2.1 from pp. 13 to 15] that $$I(m) \geq \left(I(m^2)\right)^{\ln(4/3)/\ln(13/9)}$$ which is equivalent to $$I(m^2) \leq \left(I(m)\right)^{\ln(13/9)/\ln(4/3)},$$ from which we get $$\frac{I(m^2)}{I(m)} \leq \left(I(m)\right)^{\ln(13/12)/\ln(4/3)}.$$ This last inequality implies that $$1 > \frac{I(m)}{I(m^2)} \geq \left(I(m)\right)^{\ln(4/3)/\ln(13/12)}$$ $$> \Bigg(\left(I(m^2)\right)^{\ln(4/3)/\ln(13/9)}\Bigg)^{\ln(4/3)/\ln(13/12)}= \left(I(m^2)\right)^{{\left(\ln(4/3)\right)^2}/(\ln(13/9)\cdot\ln(13/12))}.$$ But in general, we know that $I(m^2) > 8/5$, so that $$1>\frac{I(m)}{I(m^2)}> \left(I(m^2)\right)^{{\left(\ln(4/3)\right)^2}/(\ln(13/9)\cdot\ln(13/12))}$$ $$>\left(\frac{8}{5}\right)^{{\left(\ln(4/3)\right)^2}/(\ln(13/9)\cdot\ln(13/12))}> 3$$ which is a contradiction. This seems to prove that there are, in fact, no odd perfect numbers. We will stop here for the time be