On biquadratic fields: when 5 squares are not enough

In this paper we study the Pythagoras number P(OK) for the rings of integers in totally real biquadratic fields K. We continue the work of Tinková towards proving the conjecture by Krásenský, Raška and Sgallová that a biquadratic K satisfies P(OK)≥ 6 if and only if it contains neither √2 nor √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 √6 or √7 satisfy P(OK)≥ 6. Furthermore, we present ideas and computations which further support the conjecture also for K containing √3. This enables us to refine the conjecture by explicitly listing the exceptional fields.