1. If $a, b$ are integers, and $x^{2}-x-1$ is a factor of $a x^{17}$ $+b x^{16}+1$, then $a=$ $\qquad$
$=1.987$.
Let $p, q$ be the roots of the equation $x^{2}-x-1=0$, then
$$
p+q=1, pq=-1 \text{. }
$$

Given that $p, q$ are also the roots of the equation $a x^{17}+b x^{16}+1=0$, we have $\left\{\begin{array}{l}a p^{17}+b p^{16}+1=0, \\ a q^{17}+b q^{16}+1=0 .\end{array}\right.$
$$
\begin{array}{l}
\text{(1) } \times q^{16} \text{ gives} \\
a p^{17} q^{16}+b p^{16} q^{16}+q^{16}=0 . \\
\therefore a p+b=-q^{16} .
\end{array}
$$
(2) $\times p^{16}$ gives $a q+b=-p^{16}$.

Solving (3) and (4) yields
$$
\begin{aligned}
a & =(p+q)\left(p^{2}+q^{2}\right)\left(p^{4}+q^{4}\right)\left(p^{8}+q^{8}\right) \\
& =987 .
\end{aligned}
$$