Discrete Mathematics Proof Question

Question

Prove or disprove that there are infinitely many $x, y, z \in \mathbb N$ such that $$\frac{1}{x^2} + \frac{1}{y^2} = \frac{1}{z^2}$$

Currently, I tried to substitute $x, y,$ and $z$ with $2n$ and $n$ and nothing seems to work.

Answer 1

Hint Multiply your equation by $x^ny^nz^n$.

The new problem Multiplying by $x^2y^2z^2$ the new equation is

$$y^2z^2+x^2z^2=x^2y^2$$

From here it is easy to deduce that $x|yz $.

Let $yz=kx$. then the equation becomes

$$k^2+y^2=z^2 \,.$$

and $k|yz$.

If you find a solution to this equation, then you set $x= \frac{yz}{k}$.

This is easy to solve: If

$$k_0^2+y_0^2=z_0^2$$ is any solution to the pytagorean equation, then

$$(k_0^2)^2+(k_0y_0)^2=(k_0z_0)^2$$

Answer 2

We begin by noting $$\frac{1}{15^2}+\frac{1}{20^2}=\frac{1}{12^2}.$$ Multiplying both sides of this equation by $\frac{1}{k^2}$ for $k \geq 1$ gives infinitely many solutions.