Lagrange number
# Lagrange number

In mathematics, the Lagrange numbers 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.

## Definition

Hurwitz improved Peter Gustav Lejeune Dirichlet's criterion on irrationality to the statement that a real number α is irrational if and only if there are infinitely many rational numbers p/q, written in lowest terms, such that

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

This was an improvement on Dirichlet's result which had $1/q^2$ on the right hand side. The above result is best possible since the golden ratio φ is irrational but if we replace √5 by any larger number in the above expression then we will only be able to find finitely many rational numbers that satisfy the inequality for α = φ.

However, Hurwitz also showed that if we omit the number φ, and numbers derived from it, then we can increase the number √5. In fact he showed we may replace it with 2√2. Again this new bound is best possible in the new setting, but this time the number √2 is the problem. If we don't allow √2 then we can increase the number on the right hand side of the inequality from 2√2 to √221/5. Repeating this process we get an infinite sequence of numbers √5, 2√2, √221/5, ... which converge to 3. These numbers are called the Lagrange numbers, and are named after Joseph Louis Lagrange.

## Relation to Markov numbers

The nth Lagrange number $L_n$ is given by

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

where $m_n$ is the nth Markov number, that is the nth smallest integer m such that the equation

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

has a solution in positive integers x and y.

