1. ## FLT Problem

Question:

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.

2. Take $\displaystyle x = 5n;y = 3n$ then you can check that $\displaystyle w=4n$. And on the other hand we have: $\displaystyle 1 = {z^2} - 34{n^2}$ which is a Pell's Equation. (And there are infinitely many solutions)

The minimal solution is $\displaystyle (z,n)=(35,6)$ and the solutions are generated by: $\displaystyle \left\{ \begin{gathered} {z_k} = \tfrac{{{{\left( {35 + \sqrt {34} \cdot 6} \right)}^k} + {{\left( {35 - \sqrt {34} \cdot 6} \right)}^k}}} {2} \hfill \\ {n_k} = \tfrac{{{{\left( {35 + \sqrt {34} \cdot 6} \right)}^k} - {{\left( {35 - \sqrt {34} \cdot 6} \right)}^k}}} {{2\sqrt {34} }} \hfill \\ \end{gathered} \right.$

For example, the next is: $\displaystyle (z_2,n_2)=(2449,420)$

3. Thanks for your help, however, I have not yet learned about Pell's Equation in my class. Is there another way to solve this?