## a^2+b^2, a^2+c^2, b^2+c^2

$a^2 \; + \; b^2 \; = \; x^2$
$a^2 \; + \; c^2 \; = \; y^2$
$b^2 \; + \; c^2 \; = \; z^2$

Here are the parametric equations:

$a \; = \; | \:(4 \, p^2 - r^2) \, q \:|$
$b \; = \; 4 \, p \, q \, r$
$c \; = \; | \:(4 \, q^2 - r^2) \, p \:|$

where   $(p, \; q, \; r)$   is a Pythagorean triple

$p \; = \; m^2 \; - \; n^2$
$q \; = \; 2 \, m \, n$
$r \; = \; m^2 \; + \; n^2$

$a = (4 \, (m^2 - n^2)^2 - (m^2 + n^2)^2) \, (2 \, m \, n) = 6 \, m^5 \, n - 20 \, m^3 \, n^3 + 6 \, m \, n^5$

$b = 4 \,(m^2 - n^2)(2 \, m \, n) \,(m^2 + n^2) = 8 \, m^5 \, n - 8 \, m \, n^5$

$c = (4 \, (2 \, m \, n)^2 - (m^2 + n^2)^2) \, (m^2 - n^2) = -m^6 + 15 \, m^4 \, n^2 - 15 \, m^2 \, n^4 + n^6$

$a^2 + b^2$
$= (6 \, m^5 \, n - 20 \, m^3 \, n^3 + 6 \, m \, n^5)^2 + (8 \, m^5 \, n - 8 \, m \, n^5)^2$
$= 4 \, (5 \, m^5 \, n - 6 \, m^3 \, n^3 + 5 \, m \, n^5)^2$

$a^2 + c^2$
$= (6 \, m^5 \, n - 20 \, m^3 \, n^3 + 6 \, m \, n^5)^2 + (-m^6 + 15 \, m^4 \, n^2 - 15 \, m^2 \, n^4 + n^6)^2$
$= (m^2 + n^2)^6$

$b^2 + c^2$
$= (8 \, m^5 \, n - 8 \, m \, n^5)^2 + (-m^6 + 15 \, m^4 \, n^2 - 15 \, m^2 \, n^4 + n^6)^2$
$= (m^6 + 17 \, m^4 \, n^2 - 17 \, m^2 \, n^4 - n^6)^2$