Today i want to learn/discuss about of a technique for proving that the given Diophantine equation has infinitely main roots.
I just want to solve these types of problems : 1) Show that $x^2=y^3+z^5$ has infinitely many solutions for positive integers $x,y,z$. 2) Show that $x^n+y^n=z^{n-1}$ has infinitely many solutions for postive integers $x,y,z$.
To solve these types of problems a parameter is usually used which varies over integers giving infinitely many solutions.. For example, $x=k^{10}(1+k)^8 ,y=k^7(1+k)^5,z=k^4(k+1)^3$ are the solutions for problem number (1). Then as we vary $k$ over postive integers we will be getting infinitely many values.
My problem is that I am not getting motivation to how to select that values of $x,y,z$ in terms of $k$ or any parameter. After seeing the solution I feel "Ohk! It can be done using this" but i can't predict the solution. Hence I am asking is there any procedure are to be followed to solve these types of problems , any motivation inside the question?? Or it can be solved only by putting random value??
Please help me. Thanks in advance,