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## P

o

### p. 603-611

<http://www.numdam.org/item?id=JTNB_2002__14_2_603_0>

© Université Bordeaux 1, 2002, tous droits réservés.

L’accès aux archives de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/) implique l’accord avec les condi-tions générales d’utilisation (http://www.numdam.org/conditions). Toute uti-lisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte-nir la présente mention de copyright.

Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

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## folding

### fractions

par ALFRED J. VAN DER POORTEN

To Michel France on his 65th birthday

RÉSUMÉ. Le ’lemme de

### pliage’

de Michel Mendès France

four-nit une nouvelle

de la

du

### développement

en

fraction continue d’un irrationnel

ABSTRACT. Michel Mendès France’s

### ’Folding

Lemma’ for

contin-ued fraction

an unusual

### explanation

for the

well known symmetry in the

of a

irrational

### integer.

1. Continued fractions

A continued

is

al ,

Set

al , ... ,

for h =

### 0, 1, 2,

.... Then the definition

al ,

... ,

=

ao

... , ah

### immediately implies by

induction on h that there is a

between certain

of two

### by

two matrices and continued fractions. It follows

### by transposition

of matrices that

Manuscrit requ le 5 f6vrier 2001.

(3)

and

### taking

determinants that

### Proposition

2. One then sees that

It follows that if a :

if the

ai, a2, ... are

as is

the case for the

### simple

continued fraction

we discuss

and

a is

then

the

of

of the

### xh/yh

of a.

Continued fractions are well studied

so new

results

### concerning

their structure are a

3

Lemma

### [2]).

We have

Here we have set Wh =

al, a2 ... , ah;

the notation

### -w

de-notes -ah, -ah-17- -- , -al. It will save space to write a for -a.

### Proof.

Here E--~ denotes the

between two

### by

two matrices and continued fractions. We have

as

1. 0

The

is

### appropriate

because iteration of the

w, c,

a

of

on the word w

to the

### pattern

of creases in a sheet of paper

folded in

see

### Incidentally,

it is often convenient to

and to use the

convention

### whereby if,

say, y denotes yh, then

denotes the

y;

that

= yh-i .

The

### folding

lemma makes it easy,

the

of a

sum

of a

### series,

to

a continued fraction

for an

term

if the

is wide

Our

### example

is from the function field case, where

in X

the

### logarithm

of the absolute value.

(4)

of

### degree -2n

or less. Of course if the

of the

term is much

### ~ less,

that introduces a

of

As an

### application

of the lemma: the continued fraction

of the sum

is

### by

J

where the

addition of each term is done

### by

a ’fold’ with c =

see

2.

The

### computation

shows how to make

Here we use

recalled that

Thus

a

### symmetric

continued fraction

in the

### given example by

the word ah , c -

ah - 1.

This raises the

### question

as to whether every such

arises from

### folding.

Mind you, it should not raise any such

If the

the

### X/Y

and its ’first half’ is

then

### necessarily

is of the for some

c.

a better

learn from

a

sym-metric

as a folded

### expansion.

The best known case of a

sym-metric

is the

of a real

irrational

### integer.

Below

*

Jeffrey Shallit recently noticed that the evaluation just given is foreshadowed in remarks of Walter Leighton and W. T. Scott, ’A general continued fraction expansion’, Bull. Amer. Math. Soc. 45 (1939) 596-605.

(5)

3. Periodic continued fractions

the real

irrational

### integer

6 has norm n and trace t.

That

### is, b2 -

t6 -I- n = 0 with

### integers n and t, whe_re t2 -

4n &#x3E; 0 is not a

square in Z. To fix

### 6,

we suppose that 6 &#x3E;

### 6,

where 6 denotes the

of 6.

Let

al ,

be an initial

### segment

of the

continued fraction

of 6. We will

### play

with the matrices

first that the

### decomposition

defines sequences

and

of

the

+

of

Also

Because

is a

of 6 we have

and so

### Q

is at

most 6 - 3. It follows from several

of the box

that there are

X and Y

### providing

a solution to ’Pell’s

4. tXY +

= ::1:1 and

=

... , then

the matrix

### correspondencet

this is

Notice moreover that

Thus the word ao, al, ... , ar + t must be a

### palindrome~.

t There is a subtle point here at which I tacitly suppose that X 2 - tXY + ny2 = ( -1 )’’ .

However, that loses no generality, because if X/Y = [... ar also X/Y = [...

ar - 1 , 1 ). t Anubis: Dog as a devil deified lived as a god.

(6)

4.

in the

that the

of 6 is of even

r = 2h + 2. Then

N =

to half the

### period

in the sense that

## x Y

the

as in

4.

Thus we consider

whence

we have

and so,

the

### Folding

Lemma entails that

5.

that the

irrational

6

6 &#x3E;

### 3

and has a continued

even

2h + 2.

Then

and

divides 2P + t.

The

### necessity

of the condition is clear since otherwise the

### sufficiency

we need to confirm that

does

### yield

a solution to ’Pell’s

### equation’.

That is not obvious:

tXY +

=

### However,

it turns out that both

and

## QI Y.

To see

consider

the conditions

modulo

that

Px - -ny and

### ty.

Recall we suppose 2P+t - 0. Then X

= 0 and Y =

= =

as

0

that the

of 6 is of odd

### length

r = 2h + 1. Then N

to half the

### period

in the sense that

(7)

whence

and so

that

=

### Q’;

here w’ denotes the word

aI, a2,... , ah-1.

6.

that the

irrational

6 &#x3E;

### 3

and has a continued

odd

2h + 1.

Then

and

=

### Q’.

Remark. We will see below that

+ + t =

for k =

... , so

when

=

the

+ P +

is

satisfied.

The

### necessity

of the condition is clear since otherwise the

is not

in

### particular

we would not have ah =

+

P +

For its

### sufficiency

we need to confirm that does

### yield

a

solution to ’Pell’s

### equation’.

That’s not obvious:

=

both

and

## Q I Y.

Recall we have

=

and so P’ +

(8)

5. Remarks and

### explanations

5.1. Some identities. The

Px +

-ny and

+

x -

entail that

Brute

### computation

now shows that

We will avoid the need for such

that our also

the

in the form

and

=

### provides

a base from which one can prove the facts we want

### by

induction. Here we note that x =

=

### ay’ + y",

where a = ah. Then

and

these

and

y, and

That

as

As

### noted,

we therefore have

and

these

n and

and

(9)

is

### step h

in the continued fraction

of J. Just so,

fixes the

### reciprocal

of the remainder in

above and thus

and the next

~-

because the

are

it

follows from

alone that -a

for h =

### 0, 1,

.... Mind you,

we also know that the

+

### P)/Q

&#x3E; 1 and the remainders -1

0.

Given the rules

and

### defining

the continued fraction

it is an

### interesting

exercise to retrieve the conditions

and so the

decompo-sitions

of the matrices

### Nh.

The trick is to use

that

+

and

+

entails Px +

-ny

and

+

= x -

The

### point

of this all is to

that the

two

are

### confirming

that the matrices

### Nh

detail the

continued fraction

of 6.

5.3.

### Binary

forms. The matrix

determines a

form

+

+

of discriminant

+ 4Pt +

+

4n

P and

and the

and therefore

It

### might

take us a little too far afield to

### provide details,

so I leave

con-firmation of the

### following

to the interested reader. It turns out that

of a

and

of the

forms.

### Indeed,

this is a natural way to define

There

are

### provisos.

Whereas the forms determined

the

are

the

which a

### priori

has

coefficients is in

not

reduced.

it may not be

### primitive.

Its coefficients will have the

common factor

where G =

### +t).

One must therefore

consider the

matrix

5.4. The

5 and

### given

here are of course well known. I doubt however whether

### they

have heretofore been

the

Lemma.

### My

remarks on the matrices

contain

some

### previously unpublished

observations. One motivation for

### viewing

continued fractions of

irrationals as a

of the

is that

it

an immediate

to

for

### higher degree algebraic

irrationals. References

[1] F. M. DEKKING, M. MENDÈS FRANCE, A. J. VAN DER POORTEN, FOLDS!. Math. Intelligencer

4 (1982), 130-138; II: Symmetry disturbed. ibid. 173-181; III: More morphisms. ibid.

(10)

[2] M. MENDÈS FRANCE, Sur les fractions continues limitées. Acta Arith. 23 (1973), 207-215. [3] M. MENDÈS FRANCE, Principe de la symétrie perturbée. Seminar on Number Theory, Paris

1979-80, 77-98, Progr. Math. 12, Birkhäuser, Boston, Mass., 1981. [MR 83a:10089]

[4] A. J. VAN DER POORTEN, J. SHALLIT, Folded continued fractions. J. Number Theory 40

(1992), 237-250.

Alf VAN DER POORTEN

ceNTR,e for Number Theory Research

Macquarie University

Sydney, NSW 2109

Australia

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