##

Pell's Equation

The so-called ``Pell's equation'' is
with square
free, and we seek integer solutions to this equation. If
, then

Thus if are integers such that
, then
has norm , so by
Proposition 8.1.4 we have
. The integer
solutions to Pell's equation thus form a finite-index subgroup of the
group of units in the ring of integers of
. Dirichlet's
unit theorem implies that for any the solutions to Pell's equation
with not both negative forms an infinite cyclic group, which is
a fact that takes substantial work to prove using only elementary
number theory (for example, using continued fractions).
We first solve Pell's equation
with by finding
the units of the ring of integers of
using Sage.

The subgroup of cubes gives us the units with integer (not both negative).

A great article about Pell's equation is [Len02]. The
MathSciNet review begins: ``This wonderful article begins with history
and some elementary facts and proceeds to greater and greater depth
about the existence of solutions to Pell equations and then later the
algorithmic issues of finding those solutions. The cattle problem is
discussed, as are modern smooth number methods for solving Pell
equations and the algorithmic issues of representing very large
solutions in a reasonable way.''

The simplest solutions to Pell's equation can be huge, even when
is quite small. Read Lenstra's paper for some examples from
over two thousand years ago. Here is one example for
.

William Stein
2012-09-24