Integer solutions to $(a^x - b^y)/(a - b)=c$

Question

I would like to know if all integer solutions to $\frac{a^x -b^y}{a - b} = c$, where $c$ is also an integer, are known.

Answer 1

The equation is equivalent to $$a^x-b^y\equiv0\pmod{a-b}.$$ Of course $a\equiv b\pmod{a-b}$ and hence $$a^x-b^y\equiv a^x-a^y\pmod{a-b},$$ where without loss of generality $x\leq y$. So you want to find all integers $a$ and $b$ such that $a-b$ divides $a^x(1-a^y)$.

It is clear how to find all solutions; pick any $a$, $x$ and $y$ and any divisor $d$ of $a^x(1-a^y)$. Set $b=a-d$ so that $a-b$ divides $a^x(1-a^y)$ and hence also $a^x-b^y$. The above shows that every solution is of this form.