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
[edit]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
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
[edit]The nth Lagrange number Ln is given by
where mn is the nth Markov number,[3] that is the nth smallest integer m such that the equation
has a solution in positive integers x and y.
References
[edit]- Cassels, J.W.S. (1957). An introduction to Diophantine approximation. Cambridge Tracts in Mathematics and Mathematical Physics. Vol. 45. Cambridge University Press. Zbl 0077.04801.
- Conway, J.H.; Guy, R.K. (1996). The Book of Numbers. New York: Springer-Verlag. ISBN 0-387-97993-X.
External links
[edit]- Lagrange number. From MathWorld at Wolfram Research.
- Introduction to Diophantine methods irrationality and transcendence Archived 2012-02-09 at the Wayback Machine - Online lecture notes by Michel Waldschmidt, Lagrange Numbers on pp. 24–26.