Take then you can check that . And on the other hand we have: which is a Pell's Equation. (And there are infinitely many solutions)
The minimal solution is and the solutions are generated by:
For example, the next is:
Prove the following assertion: The system of simultaneous equations x^2+y^2=z^2-1 and x^2-y^2=w^2 has infinitely many solutions in positive integers x, y, z, w.
I was thinking I could consider that for any integer n>=1 and then take x=2n^2 and y=2n, but I am not sure how that would work out.