Show that there are infinitely many integer solutions for $x^2=y^3+z^5$
I can't solve this problem, i did try solving by parametric method of diophantine equations, written each variable in function of a new variable n , trying a form of envolving the variables, example,
I suppose that y=xk, such that k is integer..
however, I know that the solutions are the way: 
Best Answer
It's important to note that you are not asked to classify all solutions, just to find an infinite family of them.
Here's a simple, infinite family:
If $y=2^{5a}$, $z=2^{3a}$ then the left hand is $2^{15a+1}$ so you just need to choose $a$ to be odd in order to get a square.
For example, with $a=1$ we get $$(x,y,z)=(2^8,2^5,2^3)$$