Fermat's Last Theorem for consecutive numbers, i.e. the equation $a^n+(a+1)^n=(a+2)^n$ in natural numbers, is a well-known problem at olympiad level (it can be shown without any eliptic curves and other advanced stuff, that there are no solutions for $n\ge 3$).
Let's consider the very natural generalisation - the equation $a^x+(a+1)^y=(a+2)^z$.
I guessed the following solutions: $(a,x,y,z)$ being one of $(1,X,1,1),(1,X,3,2),(3,2,2,2)$, where $X$ can be any natural number.
Are there any others?
Can it be solved with stuff like binomial formula, modular arithmetic, estimations (olympiad level solution; these methods are sufficient to solve the mentioned case of Fermat's equation)?