If $p_n(a,b)$ is a rational number (or integer) for 3 consecutive values of $n$ then every $p_n(a,b)$ is

Question

Let $a$ and $b$ be two real numbers and $p_n(x,y)$ the polynomial: $$p_n(x,y)=\sum_{i=0}^{n-1}x^{n-1-i}y^{i},$$ where $n$ is a positive integer.

In a previous post I asked if $p_n(a,b)$ was a rational number (or an integer) for $k$ consecutive values of $n$ then $p_n(a,b)$ was a rational number (or an integer) for every $n$.

The answer is positive in both cases. If $p_n(a,b)$ is a rational for 4 consecutive values of $n$ then $p_n(a,b)$ is a rational for every $n$ by the nice answer of Vlad Matei in the previous post, and if $p_n(a,b)$ is an integer for 4 consecutive values of $n$ then $p_n(a,b)$ is an integer for every $n$, due to a problem from AMM, namely Problem E2998 by Clark Kimberling.

Do these two results remain true if instead of 4 we have only 3 consecutive values of $n$ such that $p_n(a,b)$ is a rational number (or an integer)?

Any help or reference would be appreciated.

Answer 1

It's false in general: consider $a=\sqrt[4]{2},b=-\sqrt[4]{2}$. Then $p_4(a,b)=p_6(a,b)=0,p_5(a,b)=2$ yet $p_3(a,b)=\sqrt{2}$.