Can $\left\lfloor{\dfrac{x}{2p+1}} \right\rfloor$ be expressed in terms of $\left\lfloor{\dfrac{x}{p}} \right\rfloor$ for prime $p$?
How to divide by $2p+1$ by only using division by $p$?
EDIT: The above formulation is wrong. I meant "expressed in terms" in a sense broader that "a function that takes $\left\lfloor{\dfrac{x}{p}} \right\rfloor$ as an argument.
Different version: let $0\leq a,b < 2p+1$ ($a,b$ known integers) and $x=ab$. How to divide $x$ by $2p+1$ in a way cheaper than just dividing by $2p+1$? Dividing by $p$ is cheaper than dividing by $2p+1$. It doesn't have to be a formula, algorithm is also ok.