$2 \cdot 50 N$ is an integer, its $b$-ary representation is 777, find the smallest positive integer $b$, such that $N$ is the fourth power of an integer.
[Solution] We need to find the smallest $b$ such that
$$
7 b^{2}+7 b+7=x^{4}
$$

has an integer solution for $x$. Since 7 is a prime number, 7 divides $x$, so $x=7 k$, substituting into (1) and simplifying, we get
$$
b^{2}+b+1=7^{3} k^{4} .
$$

Let $k=1$, then the above equation becomes
$$
b^{2}+b+1=7^{3} \text {, }
$$

which is $(b-18)(b+19)=0$,
so $b=18$.
Since the smallest $b$ occurs when $k$ is at its smallest, the smallest $b$ is 18.