2. (9th Canadian Mathematical Competition) $N$ is an integer, its representation in base $b$ is 777. Find the smallest positive integer $b$ such that $N$ is a fourth power of an integer in decimal notation.
2. This problem is equivalent to finding the smallest positive integer $b$, such that the equation $7 b^{2}+7 b+7=x^{4}$, (1) has an integer solution for $x$. Since 7 is a prime number, it follows from equation (1) that 7 is a divisor of $x$. Therefore, let $x=7 k$, then equation (1) becomes $b^{2}+b+1=7^{3} k^{4}$. The smallest $b$ occurs when $k$ is at its minimum. Taking $k=1$, we then have $b^{2}+b+1=343, b^{2}+b-342=0$, which is $(b-18)(b+19)=0$, yielding the positive integer solution $b=18$. Thus, we have $(777)_{18}=\left(7^{4}\right)_{10}$.