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 is irrational if and only if there are infinitely many rational numbers written in simplest terms, such thatThis was an improvement on Dirichlet's result which had on the right-hand side. The above result is best possible, since the golden ratio is irrational. If we replace with any larger number in the above expression, we will only be able to find finitely many rational numbers that satisfy the inequality for
Hurwitz also showed that if we omit, we can increase the to Again this new bound is best possible, this time with being the problem. If we omit we can further increase the to Repeating this process we get the infinite series which converges to 3. These are the Lagrange numbers, named after Joseph Louis Lagrange.
Relation to Markov numbers
The th Lagrange number is given bywhere is the th Markov number—the th-smallest integer such that the equation
has a solution in positive integers and