On biquadratic fields: when 5 squares are not enough

Abstract In this paper, we study the Pythagoras number $${\mathcal {P}}({\mathcal {O}}_K)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>K</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> for the rings of integers in totally real biquadratic fields K . We continue Tinková’s work toward proving the conjecture of Krásenský, Raška and Sgallová that a biquadratic K satisfies $${\mathcal {P}}({\mathcal {O}}_K)\ge 6$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>K</mml:mi> </mml:msub> <mml:mo>)</mml:mo> <mml:mo>≥</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> </mml:math> if and only if it contains neither $$\sqrt{2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msqrt> <mml:mn>2</mml:mn> </mml:msqrt> </mml:math> nor $$\sqrt{5}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msqrt> <mml:mn>5</mml:mn> </mml:msqrt> </mml:math> , with only finitely many exceptions. We fully resolve two of the three remaining classes of fields by proving that all but finitely many fields K containing $$\sqrt{6}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msqrt> <mml:mn>6</mml:mn> </mml:msqrt> </mml:math> or $$\sqrt{7}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msqrt> <mml:mn>7</mml:mn> </mml:msqrt> </mml:math> satisfy $${\mathcal {P}}({\mathcal {O}}_K)\ge 6$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>K</mml:mi> </mml:msub> <mml:mo>)</mml:mo> <mml:mo>≥</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> </mml:math> . Furthermore, we present ideas and computations that further support the conjecture for fields K containing $$\sqrt{3}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msqrt> <mml:mn>3</mml:mn> </mml:msqrt> </mml:math> . This enables us to refine the conjecture by explicitly listing the exceptional fields.