TITLE: Can 720! be written as the difference of two positive integer powers of 3?
QUESTION [11 upvotes]: Does the equation:
$$3^x-3^y=720!$$
have any positive integer solution?

REPLY [2 votes]: Let's write $n=x-y$.  If $720!=3^x-3^y=3^y(3^n-1)$, then $3^n-1$ is divisible by the primes $17$, $31$, $43$, and $79$ (among others, of course).  As it happens, $3$ is a primitive root for those primes.  This means that $n$ is divisible by $16$, $30$, $42$, and $78$, hence by the lcm of these, or
$$16\cdot15\cdot7\cdot39=65{,}520$$
But this is vastly larger than $\log_3(720!)\approx3660$.  Thus $720!$ is not the difference of two powers of $3$.