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](https://mathoverflow.net/questions/315532/p-nx-y-sum-i-0n-1xn-1-iyi-is-always-an-integer) 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.



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> 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)?
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Any help or reference would be appreciated.



If $p_n(a,b)$ is a rational number (or integer) for 3 consecutive values of $n$ then every $p_n(a,b)$ is
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 $p\_3(a,b)=\sqrt{2}$.


