The Lagrange number refers to two distinct entities. 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 sequence 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.
