A Proof That There Are No Odd Perfect Numbers

$$I(m) \geq \left(I(m^2)\right)^{\ln(4/3)/\ln(13/9)}$$
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 being.

Comments

  1. SilJuly 5, 2023 at 6:07 AM

    Except that $\frac{I(m^2)}{I(m)} \leq \left(I(m)\right)^{\ln(13/12)/\ln(4/3)}$ does not imply $\frac{I(m)}{I(m^2)} \geq \left(I(m)\right)^{\ln(4/3)/\ln(13/12)}$, instead $\frac{I(m)}{I(m^2)} \geq \left(I(m)\right)^{-\ln(13/12)/\ln(4/3)}$

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