Legendre's theorem on continued fractions

Builds on Convergents of a continued fraction Continued fraction expansion of a real number

Statement

Let $\alpha$ be a real number, and let $p$ and $q$ be positive integers such that $\left| \alpha - \frac{p}{q} \right| < \frac{1}{2 q^2}.$ Then $p / q$ is a convergent of the continued fraction expansion of $\alpha$ .

Remarks

So the very good rational approximations of $\alpha$ are all convergents. The constant $\frac{1}{2}$ cannot be replaced by $1$ : not every fraction with $\left| \alpha - \frac{p}{q} \right| < \frac{1}{q^2}$ is a convergent. A use: it shows that the solutions of Pell's equation come from the convergents of $\sqrt{d}$ .

History

Adrien-Marie Legendre proved the theorem in his Essai sur la théorie des nombres of 1798.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Continued fraction definition
Simple continued fraction definition
Continued fraction expansion of a real number definition
Convergents of a continued fraction definition
Legendre's theorem on continued fractions statement

Tag	`0JXTQ392ZM`
URI	`m2://0JXTQ392ZM`
Kind	statement, theorem
Writing	W0 stub
MSC	11J70
Status	This copy of the site has no build data. Run `m2 status 0JXTQ392ZM`.
Uses	Convergents of a continued fraction, Continued fraction expansion of a real number

Legendre's theorem on continued fractions

Statement

Remarks

History

Reading path

Further reading

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