On biquadratic fields: when 5 squares are not enough
Abstract
Abstract In this paper, we study the Pythagoras number $${\mathcal {P}}({\mathcal {O}}_K)$$ P ( O K ) 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 $${\mathcal {P}}({\mathcal {O}}_K)\ge 6$$ P ( O K ) ≥ 6 if and only if it contains neither $$\sqrt{2}$$ 2 nor $$\sqrt{5}$$ 5 , 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 $$\sqrt{6}$$ 6 or $$\sqrt{7}$$ 7 satisfy $${\mathcal {P}}({\mathcal {O}}_K)\ge 6$$ P ( O K ) ≥ 6 . Furthermore, we present ideas and computations that further support the conjecture for fields K containing $$\sqrt{3}$$ 3 . This enables us to refine the conjecture by explicitly listing the exceptional fields.
Fields of Study & Subjects
History and Theory of Mathematics
Algebraic Geometry and Number Theory
Advanced Differential Equations and Dynamical Systems
Publication details
- Journal / source
- Periodica Mathematica Hungarica
- Publisher
- Springer Science and Business Media LLC
- Volume
- 93
- Issue
- 1
- Pages
- 50–63
- Publication date
- 18 May 2026
- Language
- EN