## Make {x^2 + y^2 + z^2, x^2 y^2 + x^2 z^2 + y^2 z^2} squares – Part 2

Find three positive integers   $(x, y, z)$   to can make the two expressions squares

$x^2 \; + \; y^2 \; + \; z^2$
$x^2 \, y^2 \; + \; x^2 \, z^2 \; + \; y^2 \, z^2$

Here’s a parametric solution established by Euler:

$x \; = \; (n^4 - 6 \, n^2 + 1) \,(n^2 + 1)$
$y \; = \; 4 \, n \, (n^2 - 1)^2$
$z \; = \; (8 \, n^2) \,(n^2 - 1)$

$x^2 + y^2 + z^2 = (n^2+1)^6$
$x^2 \, y^2 + x^2 \, z^2 + y^2 \, z^2 = 16 \, (n-1)^2 \, n^2 \, (n+1)^2 \, (n^8-4 \, n^6+22 \, n^4-4 \, n^2+1)^2$

In two ways:

$2881585^2 \; + \; 981552^2 \; + \; 164736^2 \; = \; 145^6$
$2964815^2 \; + \; 705024^2 \; + \; 83232^2 \; = \; 145^6$