An algebra problem by Wen Z

Consider $(a+\sqrt{b})^x + (a - \sqrt{b})^x$ . For all integral $x$ , this is integral, and for all $1>a-\sqrt{b}>0$ , the fractional part of $(a+\sqrt{b})^x$ is $1-(a - \sqrt{b})^x$ (Since $1>a-\sqrt{b}>0$ and $(a+\sqrt{b})^x + (a - \sqrt{b})^x$ is integral).

Also, since $1>a-\sqrt{b}>0$ , $(a - \sqrt{b})^x$ approaches zero as $x$ increases, therefore $1-(a - \sqrt{b})^x$ approaches $1$ as $x$ increases. Therefore, we just need to find the number of integers $a,b$ such that $1>a-\sqrt{b}>0$ or $a > \sqrt{b} > a-1$ .

But, for any nonsquare $b$ , we can find a unique satisfying integer $a$ , so we just need to find the number of nonsquare numbers from $1$ to $19$ .

Our answer then becomes $\boxed{15}$ .

$\color{#FFFFFF}{Hidden Message!}$ .