July | 2022 | A Mathematician's Miscellany

It’s an interesting question how one can generalize classical continued fractions for real numbers to the p-adic setting. There are many curious p-adic continued fraction algorithms out there. I am going to demonstrate a curious construction of convergents $x_k/y_k$ whose limit is a root $i$ of $x^2 + 1$ in $\mathbb Q_p$ . It is probably an invention of a bicycle, so if you recognize where this result appeared before, let me know!

Theorem
Let $p$ be a rational prime such that $p \equiv 1 \pmod 4$ and let $i$ be a root of $x^2 + 1$ in $\mathbb Q_p$ . Let $x_1, y_1 \in \mathbb Z$ be chosen so that $x_1^2 + y_1^2 = p$ and $y_1i \equiv x_1 \pmod p$ , and define

$\begin{array}{l l}x_2 = x_1^2 - y_1^2, & y_2 = 2x_1y_1,\\x_k = (2x_1)x_{k - 1} - px_{k - 2}, & y_k = (2x_1)y_{k - 1} - py_{k - 2} \quad \text{for}\ k\geq 3.\end{array}$

Then $|y_ki - x_k|_p = p^{-k}$ for all $k \in \mathbb N$ , so the sequence $x_k/y_k$ converges to $i$ . Furthermore, for all $k \in \mathbb N$ , the rational $x_k/y_k$ has the smallest possible naive height, i.e.,