Let \(N\) be a positive integer that satisfies
\[\left(\begin{array}{cc}2 & 0 \\ 1 & 4\end{array}\right)^N\cdot \dbinom{2}{1}=\dbinom{x}{y}.\]
Find the largest possible value of \(N\) such that
\[\log_2(xy) \le 1000.\]
by
Daniel Liu
