In this paper we study the Pythagoras number $\mathcal{ P}(\mathcal{ O}_K)$ for the rings of integers in totally real biquadratic fields $K$. We continue the work of Tinkov\'a towards proving the conjecture by Kr\'asensk\'y, Ra\v{ s}ka and Sgallov\'a that a biquadratic $K$ satisfies $\mathcal{ P}(\mathcal{ O}_K)\geq 6$ if and only if it contains neither $\sqrt{ 2}$ nor $\sqrt{ 5}$, with only finitely many exceptions. We fully solve two out of three remaining classes of fields by proving that all but finitely many $K$ containing $\sqrt{ 6}$ or $\sqrt{ 7}$ satisfy $\mathcal{ P}(\mathcal{ O}_K)\geq 6$. Furthermore, we present ideas and computations which further support the conjecture also for $K$ containing $\sqrt{ 3}$. This enables us to refine the conjecture by explicitly listing the exceptional fields.
Categories: math.NT