A large prime number is $$P = 2^{32582657}-1.$$ Suppose you wish to express $P$ in the binary system used by computers, where the base is $2$ instead of $10$, and the only possible digits are $0$ and $1$. The number of binary digits is .
Hint: In the binary system, $$2^2-1 = \underline{11}, \quad\hbox{and}\quad 2^3-1 = \underline{111}$$ where the underscores indicate binary digits.