$3^x-2^y=17$ where $x$ and $y$ are both positive integers First, I was wondering what would the solution(s) be to the equation $3^x-2^y=17$ such that $x$ and $y$ are both positive integers. I couldn't find any small possibilities. Is there a proof that such a solution exists? How about the general case $3^x-2^y=z$? Is there exactly one x and y positive integer solution for every positive integer z (not divisible by 2 or 3 that is)? Thanks in advance!

It is not true that there is a solution for every odd $z$ not divisible by $3$. For example, $3^x-2^y=13$ is impossible. For we can check by hand that there is no solution if $y=1$ or $y=2$.  And if $y\ge 3$, then $13+2^y\equiv 5\pmod{8}$. But powers of $3$ are congruent to $1$ or $3$ modulo $8$.  There are infinitely many examples of this type.
We can give a partial answer to your question about the uniqueness of the solution, if there is one. It is a theorem of Stroeker and Tijdeman that the equation $3^x-2^y=z$ has at most one solution in positive integers if $|z|\gt 13$.
For more information, and references, please see OEIS.
