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This is a Quantitative Study study.

October 16, 2025Open Access

On biquadratic fields: when 5 squares are not enough

Key Points

Biquadratic fields containing neither sqrt 2 nor sqrt 5 have a pythagoras number of at least 6.
All but finitely many fields containing sqrt 6 or sqrt 7 satisfy the condition for a higher pythagoras number.
Conjecture refinement includes explicit listing of exceptional fields with specific square roots.
Ideas and computations provide support for the conjecture regarding fields containing sqrt 3.

Abstract

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.

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Daniel Dombek (Wed,) studied this question.

synapsesocial.com/papers/68f04935e559138a1a06e58e https://doi.org/https://doi.org/10.48550/arxiv.2506.20820