# Are there polynomials $P, Q$ with degree no less than 2018 and with integer coefficients, such that $P(Q(x))=3Q(P(x))+1$ for all real $x$?

Determine whether or not two polynomials $$P, Q$$ with degree no less than 2018 and with integer coefficients exist such that$$P(Q(x))=3Q(P(x))+1$$for all real numbers $$x$$.

Attempt:

I found a different construction $$P(x) = \frac{3}{2}[(2x + 1)^{n+1}-1], Q(x) = \frac{3}{2}[(2x + 1)^n-1]$$, which came from $$3x[(2x+1)^n + (2x+1)^{n-1} + ... + 1]$$. I guessed this construction from the small cases I did, and I feel that if (P, Q) works then (something, P) probably works, so I tried to inductively construct a pair of polynomials.

Any more complete and elegant solution?

• shouldn't "with degree no less than 2018" mean that all the degrees are $\ge 2018$? This means that increasing the degree doesn't lead to a contradiction Jan 7 '20 at 11:57
• @CalvinKhor I added my recent attempt Jan 7 '20 at 12:03