Perfect square

Quote:

Originally Posted by alexmahone

Prove that there are infinitely many ordered triples of positive integers $(a, b, c)$ such that $gcd\ (a, b, c) = 1$ and $a^2b^2 + b^2c^2 + c^2a^2$ is a perfect square.

put $a=b=1.$ then you only need to find infinitely many $c \in \mathbb{N}$ such that $2c^2 + 1 = x^2,$ for some $x \in \mathbb{N}.$ the equation $x^2 - 2c^2=1$ is known as Pell equation. it obviously has a solution

$x=3, \ c = 2.$ the general solution for $c$ is known to be: $c=\frac{(3+2\sqrt{2})^n - (3 - 2\sqrt{2})^n}{2\sqrt{2}}, \ \ n \in \mathbb{N}. \ \ \ \Box$

Note: it's a good exercise to prove directly, without using what is known about Pell equation, that $c_n=\frac{(3+2\sqrt{2})^n - (3 - 2\sqrt{2})^n}{2\sqrt{2}} \in \mathbb{N},$ for all $n \in \mathbb{N},$ and also that $2c_n^2 + 1$ is a perfect square.