$2^x - 3^y = k$, help understanding wolfram alpha

Question

I've been trying to understand this equation

$$2^{x} - 3^{y} = k$$

Specifically, I'm looking for all integer solutions to x and y such that k is a positive integer with $y > 1$. I figured I'd try wolfram alpha.

I suspected it would give me a family of solutions, but...

https://www.wolframalpha.com/input?i=2%5Ex+-+3%5Ey+%3D+k%2C+k%3E%3D1%2C+x%3E%3D1%2C+y%3E%3D2

It's telling me that the solutions are $$k=5, x = 5, y = 3$$ $$k=7, x = 4, y = 2$$

This particularly doesn't seem like it can be everything, especially if I say that $x = 2y$, then I have infinitely many solutions. Why is this all that wolfram alpha is telling me.

Can anyone help me interpret what's happening here and provide other resources to look at for further understanding?

Thanks!

Answer 1

Just in case people were not aware, in the case of the single exponential problem $a^{n}= t$, if there is a solution modulo all prime powers, then there is an integer solution. (A nice proof is given in Cojocaru and Murty's book "An introduction to sieve methods and their applications.")