I have confirmed: $\left\lfloor\left(\frac{3}{2}\right)^n\right\rfloor = \left\lfloor\frac{3^n-1}{2^n-1}\right\rfloor\text{ for }1< n\le500000,$ and that for $n\ge 500001,$ the chance of failure is less than $4.282484\times10^{-62470}$ percent.
Can this be useful in a proof?