Definition:Lagrange Number
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Definition
Let $\xi$ be an irrational number.
Consider a real number $L_n$ such that there are infinitely many relatively prime integers $p, q \in \Z$ such that:
- $\size {\xi - \dfrac p q} < \dfrac 1 {L_n q^2}$
We define the Lagrange number of $\xi$ to be $\map L \xi = \map \sup L$ over all $L$ satisfying the inequality above.
This article is complete as far as it goes, but it could do with expansion. In particular: According to MathWorld there are two kinds of Lagrange number. Recommend that we compare various sources for this. Also recommend that we embed this definition into some kind of mathematical context. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Also known as
This is also referred to in some sources as a Markov Constant.
Also see
Source of Name
This entry was named for Joseph Louis Lagrange.
Sources
- Weisstein, Eric W. "Lagrange Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LagrangeNumber.html
- July 2018: James Enouen: What are the Markov and Lagrange Spectra? (OSU)