Lagmental Vicfred

Quadratic Irrationals Have Periodic Continued Fractions by Vicfred

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

Lagrange's theorem characterizes real quadratic irrationals by eventual periodicity. A small computation will anchor the general statement before the abstraction takes over.

Start locally

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.

$$ \alpha\ \text{quadratic irrational} $$

The definition determines which expressions are legal; only then does the identity become meaningful. An equality in \(\mathcal A\) may change ambient meaning, so I keep \(\mathsf D\) separate from \(\mathsf C\).

$$ \alpha=[a_0;\overline{a_1,\ldots,a_\ell}]\ \text{eventually} $$

Compute before generalising

Now evaluate one representative case. The result should agree with the structural law above, but it is obtained without assuming the conclusion.

$$ \sqrt{23}=[4;\overline{1,3,1,8}] $$

The global view

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.

$$ \begin{aligned} \mathsf{D}\;&:\quad \alpha\ \text{quadratic irrational},\\[5pt] \mathsf{C}\;&:\quad \alpha=[a_0;\overline{a_1,\ldots,a_\ell}]\ \text{eventually}. \end{aligned} $$

Edge conditions

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \alpha=[a_0;\overline{a_1,\ldots,a_\ell}]\ \text{eventually} \end{gathered}} $$

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 Sun 24 June 2018. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.