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 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} ight| < 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, called the Lagrange numbers [J.H. Conway, R.K. Guy, "The Book of Numbers," New York: Springer-Verlag, pp.187-189, 1996.] , named after Joseph Louis Lagrange.

Relation to Markov numbers

The "n"th Lagrange number "Ln" is given by:L_n=sqrt{9-frac{4}m_n}^2where "mn" is the "n"th Markov number, that is the "n"th smallest integer "m" such that the equation:m^2+x^2+y^2=3mxy,has a solution in positive integers "x" and "y".

External links

* [http://mathworld.wolfram.com/LagrangeNumber.html Lagrange number] . From MathWorld at Wolfram Research.
* [http://www.math.jussieu.fr/~miw/articles/pdf/IntroductionDiophantineMethods.pdf Introduction to Diophantine methods irrationality and trancendence] - Online lecture notes by Michel Waldschmidt, Lagrange Numbers on pp.24-26.

References


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