Let $r>4$ and $n>1$ positive integers and let $α$ be a positive real number.
Let us define the following three positive integers: $$ \begin{align*} m &= \lfloor r^{(n+1)^2} \alpha \rfloor \\ j &= \lfloor r^{(n-1)^2} \alpha \rfloor \\ k &= \lfloor r^{n^2} \alpha \rfloor \end{align*} $$ where $\lfloor \cdot \rfloor$ is the floor function.
My question is: Can we find a relation between the three integers $m$, $j$ and $k$?
I know that $j<k<m$. I want to see relations of the form: $m=bk,k=cj$,..etc.
This is impossible because m, j, and k are all not linearly related. Consider this definition: $$ m=r^x$$$$ j=r^y$$$$ k=r^z $$ Since $x \neq y \neq z$, these variables can't be related through multiplication by a constant. This is equivalent to trying to represent a quadratic equation as a linear equation, which is clearly impossible.