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We study real quadratic fields $\mathbb{Q}(\sqrt{D})$ such that, for a givenrational integer $m$, all $m$-multiples of totally positive integers are sumsof squares. We prove quite sharp necessary and sufficient conditions for thisto happen. Further, we give a fast algorithm that solves this question forspecific $m$, $D$ and we give complete results for $m \leq 5000$.

Representing multiples of $m$ in real quadratic fields as sums of squares