Tunnell's congruent number conjecture criterion

Let $n$ be a positive square-free integer. Set $a=1$ if $2 mid n$ and $a=2$ otherwise, and let $\Sigma(n)$ be the set of integral solutions of

$$2ax^2+y^2+8z^2=\frac{n}{a}.$$

Tunnell's conjecture. The integer $n$ is a congruent number if and only if

$$\#\{(x,y,z)\in\Sigma(n):2\mid z\}=\#\{(x,y,z)\in\Sigma(n):2\nmid z\}.$$

This is the congruent-number criterion arising from the Birch and Swinnerton-Dyer conjecture and Tunnell's theorem; the source presents it as an equivalence without stating its resolution status.