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

The minimal solution is $(z,n)=(35,6)$ and the solutions are generated by: $\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: $(z_2,n_2)=(2449,420)$

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?