2. $50 N$ is an integer, its base $b$ representation is 777, find the smallest positive integer $b$, such that $N$ is an integer to the fourth power.
[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
$$b^{2}+b+1=7^{3}$$

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.