Existence of Positive Integer Solution for a Polynomial Equation

Question

Let $f(x)$ be a polynomial of degree n, and it is known that $f(x)$ has no positive integers as its roots. In other words, there are no positive integer values of x for which $f(x)=0$. Consider the equation: $f(x+1)=2f(x)$ I am interested in determining whether there exists a solution in the positive integer set for this equation. Can someone provide insight into whether a positive integer solution x exists that satisfies this equation? Additionally, could you please explain the reasoning behind your answer? Thank you in advance for your help and expertise.

Answer 1

Let $f(x)=x^2+2$. It has no integer roots. However, for $x=1$ we have $$ f(x+1)=f(2)=6=2f(1)=2f(x). $$