Lagrange number

From Wikipedia, the free encyclopedia

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.

1 Definition
2 Relation to Markov numbers
3 External Links
4 References

[edit] Definition

Hurwitz improved 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/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, called the Lagrange numbers^[1], named after Joseph Louis Lagrange.