Lagrange number

In mathematics, the Lagrange numbers (A382098 and A382099 in the OEIS) are a sequence of numbers that appear in bounds relating to the approximation of irrational numbers by rational numbers. They are linked to Hurwitz's theorem.

Hurwitz improved Peter Gustav Lejeune Dirichlet's criterion on irrationality to the statement that a real number ${\displaystyle \alpha }$ is irrational if and only if there are infinitely many rational numbers ${\textstyle {\frac {p}{q}}}$, written in simplest terms, such that

${\displaystyle \left|\alpha -{\frac {p}{q}}\right|<{\frac {1}{{\sqrt {5}}q^{2}}}}$.

This was an improvement on Dirichlet's result which had ${\textstyle {\frac {1}{q^{2}}}}$ on the right-hand side. The above result is best possible, since the golden ratio ${\displaystyle \varphi }$ is irrational. If we replace ${\displaystyle {\sqrt {5}}}$ with any larger number in the above expression, we will only be able to find finitely many rational numbers that satisfy the inequality for ${\displaystyle \alpha =\varphi }$.

Hurwitz also showed that if we omit ${\displaystyle \varphi }$ (and numbers derived therefrom), we can increase the ${\displaystyle {\sqrt {5}}}$ to ${\displaystyle 2{\sqrt {2}}}$. Again this new bound is best possible, this time with ${\displaystyle {\sqrt {2}}}$ being the problem. If we omit${\displaystyle {\sqrt {2}}}$, we can further increase the ${\displaystyle 2{\sqrt {2}}}$ to ${\textstyle {\frac {\sqrt {221}}{5}}}$. Repeating this process we get the infinite series ${\textstyle {\sqrt {5}},\;2{\sqrt {2}},\;{\frac {\sqrt {221}}{5}},\;{\frac {\sqrt {1517}}{13}},\;\ldots }$ which converges to 3.¹ These are the Lagrange numbers,² named after Joseph Louis Lagrange.

The ${\displaystyle n}$th Lagrange number ${\displaystyle L_{n}}$ is given by

${\displaystyle L_{n}={\sqrt {9-{\frac {4}{M_{n}^{2}}}}}}$

where ${\displaystyle M_{n}}$ is the ${\displaystyle n}$th Markov number³—the ${\displaystyle n}$th-smallest integer ${\displaystyle m}$ such that the equation

${\displaystyle m^{2}+x^{2}+y^{2}=3mxy}$

has a solution in positive integers ${\displaystyle x}$ and ${\displaystyle y}$.