For example, $5 n$ is a positive integer, its base $b$ representation is 777, find the smallest positive integer $b$ such that $n$ is a fourth power of an integer.
Solution: Let $b$ be the smallest positive integer, then $7 b^{2}+7 b+7=x^{4}\left(x \in Z^{+}\right)$, which means the prime number 7 can divide $x^{4}$, hence 7 can divide $x$.
Let $x=7 m, m \in Z^{+}$, then $b^{2}+b+1=7^{3} m^{4}$.
When $m=1$, the equation becomes $(b-18)(b+19)=0$, thus $b=18$ is the smallest positive integer sought.