Pell equation
A Pell equation is a type of diophantine equation in the form for a natural number
. Generally,
is taken to be square-free, since otherwise we can "absorb" the largest square factor
into
by setting
.
Notice that if is a perfect square, then this problem can be solved using difference of squares. We would have
, from which we can use casework to quickly determine the solutions.
Alternatively, if D is a nonsquare then there are infinitely many distinct solutions to the pell equation. To prove this it must first be shown that there is a single solution to the pell equation.
Claim: If D is a positive integer that is not a perfect square, then the equation x^2-Dy^2 = 1 has a solution in positive integers.
Proof: Let be an integer greater than 1. We will show that there exists integers
and
such that
with
. Consider the sequence
. By the pigeon hole principle it can be seen that there exists i, j, and p such that i < j,
and
.
So we now have
.
We can now create a sequence of such that
and
which implies
r and s. However we can see by the pigeon hole principle that there is another infinite sequence which will be denoted by
such that
. Once again, from the pigeon hole principle we can see that there exist integers f and g such that
mod H,
mod H, and
. Define
and notice that
. Also note that
mod H which means that Y = 0 mod H also. We can now see that
is a nontrivial solution to pell's equation.
Family of solutions
Let be the minimal solution to the equation
. Note that if
are solutions to this equation then
which means
is another solution. From this we can guess that
is obtained from
. This does indeed generate all the solutions to this equation. Assume there was another solution
. Then there exists some m such that
.
However, it can be seen that
Meaning is a solution smaller than the minimal solution which is a contradiction.
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Continued fractions
The solutions to the Pell equation when is not a perfect square are connected to the continued fraction expansion of
. If
is the period of the continued fraction and
is the
th convergent, all solutions to the Pell equation are in the form
for positive integer
.
Generalization
A Pell-like equation is a diophantine equation of the form , where
is a natural number and
is an integer.