Find all positive integers satisfying: $x^5+y^6=z^7$ 
Find all positive integers satisfying: $x^5+y^6=z^7$

No algebraic method came into my mind,just tried to find some answers and failed! Of course it's very simple to write a computer program finding at least one solution but I prefer not to use computer.
To me it's like Fermat's equation!!!
So hard!!

Partial answer
One family of solutions.
Set $x=2^{6a},y=2^{5a}$ and $z=2^{b}$
So, it's sufficient to prove $30a+1=7b$ has infinite solutions, which is left as an exercise to the reader.

 Observe that $30 \cdot 3+1=7 \cdot 13$
 Now set $a=7k+3$ and $b=30k+13$, where $k \in \mathbb{N}$ to get the required result.

