# 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

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

This was an improvement on Dirichlet's result which had 1/q2 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.[1] These numbers are called the Lagrange numbers,[2] and are named after Joseph Louis Lagrange.

## Relation to Markov numbers

The nth Lagrange number Ln is given by

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

where mn is the nth Markov number,[3] that is the nth smallest integer m such that the equation

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

has a solution in positive integers x and y.

## References

1. ^ Cassels (1957) p.14
2. ^ Conway&Guy (1996) pp.187-189
3. ^ Cassels (1957) p.41