Binary Perfection

Let $q=82589933$ . All known perfect numbers are of the form $2^{p-1}(2^p-1)$ where $2^p-1$ is a Mersenne prime. Furthermore, all Mersenne primes correspond to a perfect number of this form. Thus, the $51$ st known perfect number is $2^{q-1}(2^q-1) = 2^{2q-1}-2^q$ . Note that $2^q<2^{2q-2}$ . Thus, $2^{2q-2}<2^{2q-1}-2^q<2^{2q-1}$ and $2^{2q-1}-2^q$ has $2q-1 = \boxed{165179865}$ binary digits.