How to find polynomial with rational coefficients, which is congruent to given function (or prove that it doesn't exist)?
Particularly, how to find $P(x)$, so $$P(x)\equiv 4\left\lfloor\frac{x}{6}\right\rfloor\pmod{6}, x\in\mathbb{Z}$$?
I know that in some cases it's possible, for example, I have found:
$$\frac{2}{3}x(x-1)(x-2)\equiv 4\left\lfloor\frac{x}{3}\right\rfloor\pmod{6}, x\in\mathbb{Z}$$
I tried similar approach like in example, but polynomials like this don't work:
$$\text{const}\cdot x(x-1)(x-2)(x-3)(x-4)(x-5)$$
I also tried to create system of equations to find coefficients of polynomial with fixed degree. However, only some first values were congruent to given function.