Definition 2.1
A
periodic continued fraction is a continued
fraction
such that
for a fixed positive integer
and all sufficiently large
.
We call
the
period of the continued fraction.
Proof.
(
) First suppose that
is a periodic continued fraction. Set
. Then
so
(We use that
is the last partial convergent.)
Thus
satisfies a quadratic equation. Since the
are
all integers, the number
can be expressed as a polynomial in
with
rational coefficients, so
also satisfies a quadratic polynomial.
Finally,
because periodic continued fractions
have infinitely many terms.
(
)
This direction was first proved by Lagrange. The proof
is much more exciting!
Suppose
satisfies a quadratic equation
with
.
Let
be the expansion of
. For each
, let
so that
We have
Substituting this expression for
into the quadratic equation
for
, we see that
where
Note that
, that
, and that
Recall from the proof of Theorem 2.3 of the previous lecture that
Thus
so
with
Hence
Thus
Thus there are only finitely many possibilities for the integer
.
Also,
and
so there are only finitely many triples
,
and hence only finitely many possibilities for
as
varies. Thus for some
,
This shows that the continued fraction for
is periodic.