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

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.

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 $$p_3(a,b)=\sqrt{2}$$.