Division-line conjecture for Pythagoras numbers of integers in biquadratic fields

Let $K$ be a totally real biquadratic field, and write $\mathcal{P}(\mathcal{O}_K)$ for the least number of squares of algebraic integers needed to represent every totally positive algebraic integer of $K$. The fields $K_{m,s}$ are the biquadratic fields used in the paper's notation.

Division-line conjecture.

$$\mathcal{P}(\mathcal{O}_K)\leq 5$$

if and only if at least one of the following holds:

• $K$ contains $\sqrt{2}$ or $\sqrt{5}$;
• $K=K_{3,s}$ with $s\in\{7,10,11,13,14\}$;
• $K=K_{6,14}$.

The paper proves the lower bound $\mathcal{P}(\mathcal{O}_K)\geq 6$ in broad classes and gives substantial computational evidence, but the upper bound for the six exceptional fields in the conjecture remains open.