Sums of nonvanishing integral squares in real quadratic fields

In 1911, Dubois determined all positive integers that are represented by sums of k nonvanishing squares for any k≥4. In this article, we extend the Dubouis' results to real quadratic fields Q(m) and we will show that, for each positive integer k≥5, there exists a bound C(m,k) such that every totally positive integer in the real quadratic field Q(m) whose norm exceeds C(m,k) can be expressed as a sum of k nonvanishing integral squares in Q(m).