Let

\begin{align*} Q: \mathbb{Z}^3 &\to \mathbb{Z} \end{align*}

be a homogeneous quadratic form with integer coefficients, and let $D,N \in \mathbb{Z}$ with $D$ not a square. The following decision procedure consists of sound reductions and solution-preserving transformations for equations of the forms $Q(x,y,z)=0$ and $x^2-Dy^2=N$.

1. If an integral solution exists, then its reduction modulo every integer $m \geq 2$, in particular modulo small primes and modulo $8$, gives a solution of the corresponding congruence. 2. If the projective conic

\begin{align*} C_Q(\mathbb{Q}) := \{[x:y:z] \in \mathbb{P}^2(\mathbb{Q}) : Q(x,y,z)=0\} \end{align*}

contains a rational point $P$, then every rational point of $C_Q$ lies on a rational line through $P$, and the remaining intersection of such lines with $C_Q$ gives the usual rational parametrisation, except for the tangent-line degeneracy.

3. The equation $x^2-Dy^2=N$ is the norm equation

\begin{align*} N_{K/\mathbb{Q}}(x+y\sqrt{D})=N \end{align*}

in the quadratic field $K=\mathbb{Q}(\sqrt{D})$, with the full ring of integers $\mathcal{O}_K$ used when integrality requires it.

4. If $\varepsilon \in \mathcal{O}_K^\times$ is a unit of norm $1$, then multiplication by powers of $\varepsilon$ sends solutions of the norm equation to further solutions of the same norm equation. 5. A descent argument is valid whenever a hypothetical nonzero solution can be assigned a positive integer complexity and transformed into another solution of strictly smaller positive complexity.