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

TL;DR

This paper investigates conditions under which multiples of integers in real quadratic fields can be expressed as sums of squares, providing sharp criteria, an efficient algorithm, and comprehensive results for multiples up to 5000.

Contribution

It establishes precise necessary and sufficient conditions for representing multiples as sums of squares in real quadratic fields and offers a fast algorithm for specific cases.

Findings

Sharp necessary and sufficient conditions derived

Algorithm developed for specific m, D

Complete results for m ≤ 5000

Abstract

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