periodic continued fractions represent quadratic irrationals

periodic continued fractions represent quadratic irrationals

This article shows that

infinite

simple continued fractions

that are

eventually periodic

correspond precisely to quadratic

irrationals

.

Throughout, we will freely use results on

convergents

to a

continued fraction

; see that article for details.

Definition 1

.

A

periodic

simple continued fraction

is a simple continued fraction

[

a

0

,

a

1

,

a

2

,

…

]

such that for some

k

≥

0

there is

m

>

0

such that whenever

r

≥

k

, we have

a

r

=

a

r

+

m

. Informally, a

periodic continued fraction

is one that

eventually

repeats. A

purely periodic

simple continued fraction is one for which

k

=

0

; that is, one whose repeating period starts with the initial element.

If

[

a

0

,

a

1

,

…

,

a

k

-

1

,

a

k

,

…

,

a

k

+

j

-

1

,

a

k

,

…

,

a

k

+

j

-

1

,

a

k

,

…

]

is a periodic continued fraction, we write it as

[

a

0

,

a

1

,

…

,

a

k

-

1

,

a

k

,

…

,

a

k

+

j

-

1

¯

]

.

Theorem 1

.

If

α

=

[

a

0

,

a

1

,

…

,

a

r

,

b

1

,

…

,

b

t

¯

]

is a periodic simple continued fraction, then

α

is a quadratic irrational

p

+

q

⁢

d

for

p

,

q

rational

and

d

squarefree

. Conversely, every such quadratic irrational is represented by such a continued fraction.

Proof.

The forward direction is pretty straightforward. Given such a continued fraction, let

β

be the

(

r

+

1

)

st

complete convergent, i.e.

β

=

[

b

1

,

…

,

b

t

¯

]

Note first that

β

must be irrational since the continued fraction for any rational number terminates. Then the article on convergents to a continued fraction shows that

β

=

β

⁢

p

t

+

p

t

-

1

β

⁢

q

t

+

q

t

-

1

where the

p

i

,

q

i

are the convergents to the continued fraction for

β

. Thus

q

t

⁢

β

2

+

(

q

t

-

1

-

p

t

)

⁢

β

-

p

t

-

1

=

0

and thus

β

is irrational and satisfies a

quadratic equation

so is a quadratic irrational. A simple computation then shows that

α

is as well.

In the other direction, suppose that

α

is a quadratic irrational satisfying

a

⁢

α

2

+

b

⁢

α

+

c

=

0

and with continued fraction

representation

α

=

[

a

0

,

a

1

,

…

]

Then for any

n

>

0

, we have

α

=

t

n

⁢

p

n

-

1

+

p

n

-

2

t

n

⁢

q

n

-

1

+

q

n

-

2

where the

t

i

are the complete convergents of the continued fraction, so that from the quadratic equation we have

A

n

⁢

t

n

2

+

B

n

⁢

t

n

+

C

n

=

0

where

A

n

=

a

⁢

p

n

-

1

2

+

b

⁢

p

n

-

1

⁢

q

n

-

1

+

c

⁢

q

n

-

1

2

B

n

=

2

⁢

a

⁢

p

n

-

1

⁢

p

n

-

2

+

b

⁢

(

p

n

-

1

⁢

q

n

-

2

+

p

n

-

2

⁢

q

n

-

1

)

+

2

⁢

c

⁢

q

n

-

1

⁢

q

n

-

2

C

n

=

a

⁢

p

n

-

2

2

+

b

⁢

p

n

-

2

⁢

q

n

-

2

+

c

⁢

q

n

-

2

2

=

A

n

-

1

Note that

A

n

≠

0

for each

n

>

0

since otherwise

a

⁢

p

n

-

1

2

+

b

⁢

p

n

-

1

⁢

q

n

-

1

+

c

⁢

q

n

-

1

2

=

0

so that

a

⁢

(

p

n

-

1

q

n

-

1

)

2

+

b

⁢

p

n

-

1

q

n

-

1

+

c

=

0

and the quadratic equation would have a rational root, contradicting the fact that

α

is irrational.

The

remainder

of the proof is an elaborate computation that shows we can bound each of

A

n

,

B

n

,

C

n

independent of

n

. Assuming that, it follows that there are only a finite number of possibilities for the triples

(

A

n

,

B

n

,

C

n

)

, so we can choose

n

1

,

n

2

,

n

3

such that

(

A

n

1

,

B

n

1

,

C

n

1

)

=

(

A

n

2

,

B

n

2

,

C

n

2

)

=

(

A

n

3

,

B

n

3

,

C

n

3

)

Then each of

t

n

1

,

t

n

2

,

t

n

3

is a root of (say)

A

n

1

⁢

t

2

+

B

n

1

⁢

t

+

C

n

1

so that two of them must be equal. But then

t

n

1

=

t

n

2

(say), and

t

n

1

=

[

a

n

1

,

a

n

1

+

1

,

…

]

t

n

2

=

[

a

n

2

,

a

n

2

+

1

,

…

]

and the continued fraction is periodic.

We proceed to find the bounds. We know that

|

α

-

p

n

-

1

q

n

-

1

|

<

1

q

n

-

1

2

so that

α

-

p

n

-

1

q

n

-

1

=

ϵ

q

n

-

1

2

for some

ϵ

(depending on

n

) with

|

ϵ

|

<

1

. Thus

A

n

=

a

⁢

p

n

-

1

2

+

b

⁢

p

n

-

1

⁢

q

n

-

1

+

c

⁢

q

n

-

1

2

=

a

⁢

(

α

⁢

q

n

-

1

+

ϵ

q

n

-

1

)

2

+

b

⁢

q

n

-

1

⁢

(

α

⁢

q

n

-

1

+

ϵ

q

n

-

1

)

+

c

⁢

q

n

-

1

2

=

(

a

⁢

α

2

+

b

⁢

α

+

c

)

⁢

q

n

-

1

2

+

2

⁢

a

⁢

α

⁢

ϵ

+

a

⁢

ϵ

2

q

n

-

1

2

+

b

⁢

ϵ

=

2

⁢

a

⁢

α

⁢

ϵ

+

a

⁢

ϵ

2

q

n

-

1

2

+

b

⁢

ϵ

so that

|

A

n

|

=

|

2

⁢

a

⁢

α

⁢

ϵ

+

a

⁢

ϵ

2

q

n

-

1

2

+

b

⁢

ϵ

|

<

2

⁢

|

a

⁢

α

|

+

|

a

|

+

|

b

|

and thus also

|

C

n

|

<

2

⁢

|

a

⁢

α

|

+

|

a

|

+

|

b

|

It remains to bound

B

n

. But

B

n

2

-

4

⁢

A

n

⁢

C

n

=

(

2

⁢

A

n

⁢

t

n

+

B

n

)

2

Substituting the values of

A

n

,

B

n

on the right, and using the fact that

t

n

=

-

p

n

-

2

-

x

⁢

q

n

-

2

p

n

-

1

-

x

⁢

q

n

-

1

we get after a computation

B

n

2

-

4

⁢

A

n

⁢

C

n

=

(

p

n

-

1

⁢

q

n

-

2

-

p

n

-

2

⁢

q

n

-

1

)

⁢

2

⁢

a

⁢

x

⁢

p

n

-

1

+

b

⁢

p

n

-

1

+

b

⁢

x

⁢

q

n

-

1

+

2

⁢

c

⁢

q

n

-

1

p

n

-

1

-

x

⁢

q

n

-

1

=

±

(

2

⁢

a

⁢

x

+

b

)

⁢

p

n

-

1

+

(

2

⁢

a

⁢

x

2

+

2

⁢

b

⁢

x

+

2

⁢

c

)

⁢

q

n

-

1

-

(

2

⁢

a

⁢

x

2

+

b

⁢

x

)

⁢

q

n

-

1

p

n

-

1

-

x

⁢

q

n

-

1

=

±

(

p

n

-

1

-

x

⁢

q

n

-

1

)

⁢

(

2

⁢

a

⁢

x

+

b

)

p

n

-

1

-

x

⁢

q

n

-

1

=

±

(

2

⁢

a

⁢

x

+

b

)

so that

B

n

2

-

4

⁢

A

n

⁢

C

n

=

(

2

⁢

a

⁢

x

+

b

)

2

=

b

2

-

4

⁢

a

⁢

c

Thus

B

n

2

≤

4

⁢

|

A

n

⁢

C

n

|

+

|

b

2

-

4

⁢

a

⁢

c

|

<

4

⁢

(

2

⁢

|

a

⁢

α

|

+

|

a

|

+

|

b

|

)

2

+

|

b

2

-

4

⁢

a

⁢

c

|

and we have thus

bounded

all of

A

n

,

B

n

,

C

n

independent of

n

.
∎

References

1

G.H. Hardy & E.M. Wright,

An Introduction to the Theory of Numbers

, Fifth Edition, Oxford Science Publications, 1979.

Title

periodic continued fractions represent quadratic irrationals

Canonical name

PeriodicContinuedFractionsRepresentQuadraticIrrationals

Date of creation

2013-03-22 18:04:40

Last modified on

2013-03-22 18:04:40

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

9

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 11Y65

Classification

msc 11A55

Defines

periodic continued fraction

Defines

purely periodic

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