  We establish some upper bounds for the number of integer solutions to the
Thue inequality $|F(x , y)| \leq m$, where $F$ is a binary form of degree $n
\geq 3$ and with non-zero discriminant $D$, and $m$ is an integer. Our upper
bounds are independent of $m$, when $m$ is smaller than
$|D|^{\frac{1}{4(n-1)}}$. We also consider the Thue equation $|F(x , y)| = m$
and give some upper bounds for the number of its integral solutions. In the
case of equation, our upper bounds will be independent of integer $m$, when $ m
< |D|^{\frac{1}{2(n-1)}}$.
