Lagmental Vicfred

Convergents Satisfy a Determinant Identity by Vicfred

Last updated: Sun 15 January 2023

tags: mathematics, continued fractions, Diophantine approximation, Pell equations

Neighboring convergents are reduced and differ by the reciprocal of a product of denominators. A small computation will anchor the general statement before the abstraction takes over.

Definitions first

A continued fraction \([a_0;a_1,a_2,\ldots]\) produces convergents \(p_n/q_n\) with exceptional rational approximation. Quadratic irrationals are exactly the eventually periodic cases.

$$ p_n=a_np_{n-1}+p_{n-2},\qquad q_n=a_nq_{n-1}+q_{n-2} $$

The first display fixes the mathematical data. I label it \(\mathsf{data}\) mentally, while the next is the \(\mathsf{claim}\); the bridge between them is the displayed \(\Longrightarrow\), not an automatic implication.

$$ p_nq_{n-1}-p_{n-1}q_n=(-1)^{n-1} $$

A small case in full

Here is a concrete symbolic test. Reading it from left to right reveals which transformation is reversible and which is only an implication.

$$ \left|\frac{p_n}{q_n}-\frac{p_{n-1}}{q_{n-1}}\right|=\frac1{q_nq_{n-1}} $$

The reusable statement

What survives the example is not its particular numbers but the relation encoded by the two rows below. That relation is the part worth transporting to a new setting.

$$ \begin{aligned} \mathsf{D}\;&:\quad p_n=a_np_{n-1}+p_{n-2},\qquad q_n=a_nq_{n-1}+q_{n-2},\\[5pt] \mathsf{C}\;&:\quad p_nq_{n-1}-p_{n-1}q_n=(-1)^{n-1}. \end{aligned} $$

A nearby false statement

Good approximation does not mean arbitrary denominator. The convergents are special because their determinants alternate between plus and minus one.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] p_nq_{n-1}-p_{n-1}q_n=(-1)^{n-1} \end{gathered}} $$

With the dependency made explicit, the same pattern can be recognised safely in nearby problems. A changed hypothesis should now be easy to spot.

This article was posted on Tue 29 October 2019. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.