Let $p$ be an odd prime. Does the equation
$$2^x-3p^y=5$$
only have finitely many solutions in positive integers $x$ and $y$?



Does $2^x-3p^y=5$ (with $p$ an odd prime) have only finitely many positive integer solutions?
The solutions of your equation can be injected into the solutions of the $S$-unit equation over $\mathbb{Q}$, where $S=\{\infty,2,3,5,p\}$. As the latter is known to have finitely many solutions by the results of Siegel, Mahler, Lang (see Chapter 5 in Bombieri-Gubler: Heights in Diophantine geometry), your equation also has finitely many solutions.


By a [result of Beukers and Schlickewei](http://matwbn.icm.edu.pl/ksiazki/aa/aa78/aa7826.pdf) (see Theorem 5.2.1 in the above mentioned book), the number of solutions is at most $2^{72}$ for any given prime $p$. In fact the earlier [work of Evertse](http://www.digizeitschriften.de/download/PPN356556735_0075/PPN356556735_0075___log37.pdf) (Inventiones, 1984) yields the better bound $3\cdot7^{7}$ for the number of solutions (apply Theorem 1 with $\lambda=1/5$, $\mu=3/5$, $S=\{\infty,2,p\}$).


