Find all the solutions $(x,y,z)$ $\in \Bbb Z^+$ such that $(x+1)^y-x^z=1$.
This is an old problem from a math olympiad in Venezuela, in the year 2000.
I don't know how to start solving this kind of problem, and i didn't found any solution yet in internet, i only found the problem.
The only thing i tried was to manipulate the equation: $$(x+1)^y-x^z=1$$ $$(x+1)^y=1+x^z$$
and after that maybe use the Binomial theorem, but i think it doesn't work.
Edit: It's trivial that $y=z=1$ makes infinite solutions for any value of $x$, but what happens to $x,y,z \gt 1$ ?
Any hints?.