In number theory, specifically in Diophantine approximation theory, the Markov constant

is the factor for which Dirichlet's approximation theorem can be improved for

Certain numbers can be approximated well by certain rationals; specifically, the convergents of the continued fraction are the best approximations by rational numbers having denominators less than a certain bound.

Dirichlet proved in 1840 that the least readily approximable numbers are the rational numbers, in the sense that for every irrational number there exists infinitely many rational numbers approximating it to a certain degree of accuracy that only finitely many such rational approximations exist for rational numbers.

Specifically, he proved that for any number

there are infinitely many pairs of relatively prime numbers

51 years later, Hurwitz further improved Dirichlet's approximation theorem by a factor of √5,[2] improving the right-hand side from

for irrational numbers: The above result is best possible since the golden ratio

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

Again this new bound is best possible in the new setting, but this time the number √2, and numbers equivalent to it, limits the bound.

However, instead of considering Hurwitz's theorem (and the extensions mentioned above) as a property of the real numbers except certain special numbers, we can consider it as a property of each excluded number.

Thus, the theorem can be interpreted as "numbers equivalent to

are among the least readily approximable irrational numbers."

This leads us to consider how accurately each number can be approximated by rationals - specifically, by how much can the factor in Dirichlet's approximation theorem be increased to from 1 for that specific number.

Mathematically, the Markov constant of irrational

[4] If the set does not have an upper bound we define

is defined as the closest integer to

Hurwitz's theorem implies that

is its continued fraction expansion then

is a quadratic irrational number.

In fact, the lower bound for

are families of quadratic irrationalities having the same period (but at different offsets), and the values of

, no two of which have the same ending; for instance, for each number

forms the Lagrange spectrum.

Burger et al. (2002)[8] provides a formula for which the quadratic irrationality

whose Markov constant is the nth Lagrange number:

is the nth Markov number, and u is the smallest positive integer such that

Nicholls (1978)[9] provides a geometric proof of this (based on circles tangent to each other), providing a method that these numbers can be systematically found.

because the continued fraction representation of e is unbounded.

By trial and error it can be found that