= Unit reduction for quadratic norm equations
{title2=$\alpha\mapsto\alpha\varepsilon^n$}
In a real quadratic integer ring, multiplying by a positive norm-one fundamental unit preserves a prescribed norm. Choose an exponent so that a positive representative lies in a half-open interval $[A,A\varepsilon)$. Its conjugate is the prescribed norm divided by the representative, so both integer coefficients are bounded on that interval. Enumerating these reduced representatives gives a complete finite list of unit orbits of solutions.
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