There are two distinct entities both known as the Lagrange number. The more common one arises in rational approximation theory (Conway and Guy 1996), while the other refers to solutions of a particular Diophantine equation (Dörrie 1965).
Hurwitz's irrational number theorem gives the best rational approximation possible
for an arbitrary irrational number 
as
 |
(1)
|
The 
are called Lagrange numbers, and get steadily larger for each "bad" set
of irrational numbers which is excluded, as indicated in the following table.
 | exclude |  |
| 1 | none |  |
| 2 |  |  |
| 3 |  |  |
Lagrange numbers are of the form
 |
(2)
|
where 
is a Markov number. The Lagrange numbers form a
spectrum called the Lagrange
spectrum.
Given a Pell equation (a quadratic Diophantine equation)
 |
(3)
|
with 
a quadratic surd, define
 |
(4)
|
for each solution with 
.
The numbers 
are then known as Lagrange numbers (Dörrie 1965). The product and quotient of
two Lagrange numbers are also Lagrange numbers. Furthermore, every Lagrange number
is a power of the smallest Lagrange number with an integer
exponent.
