The diophantine equation $ax^2+bxy+cy^2=N$, $D=b^2-4ac>0$

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

K

EITH

M

ATTHEWS

The diophantine equation ax

2

+ bxy + cy

2

= N,

D

= b

2

− 4ac > 0

Journal de Théorie des Nombres de Bordeaux, tome

14, n

o

_{1 (2002),}

p. 257-270

<http://www.numdam.org/item?id=JTNB_2002__14_1_257_0>

© Université Bordeaux 1, 2002, tous droits réservés.

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

The

_{Diophantine equation}

ax2

+

_bxy

+

cy2 = N,

D = b2 -

4ac

&#x3E;

0

par KEITH MATTHEWS

RÉSUMÉ. Nous revisitons un

_algorithme

dû à

_Lagrange,

basé sur le

_{développement}

en fraction _{continue, pour} résoudre

_{l’équation}

ax2 +

_bxy

+

_cy2

= N en les entiers _{x, y}

_premiers

_{entre eux,} où

N ~

0,

pgcd(a,

b,

c)

=

pgcd(a, N)

=1 et D = b2 - 4ac &#x3E; 0 n’est

pas un carré.

ABSTRACT. We make more accessible a

neglected simple

contin-ued fraction based

_algorithm

due to

_Lagrange,

for

_deciding

the

solubility

of ax2 +

_bxy

+

cy2

= N in

_relatively

_prime

_integers

_x,

y,

where

_{N ~}

_0,

_gcd(a,

_b,

_c)

=

gcd(a, N)

=1 and D = b2 - 4ac _&#x3E; 0 is

not a

_perfect

square. In the case of

_solubility,

solutions with least positive y, from each

equivalence class,

are also constructed.

Our paper is a

generalisation

of an earlier paper

by

the author

on the

equation x2 - Dy2

= N. As in that _{paper, we use a}

lemma on unimodular matrices that

_gives

a much

simpler proof

than

_Lagrange’s

for the

_necessity

of the existence of a solution.

Lagrange

did not discuss an

exceptional

case which can arise when D = 5. This _was done

_by

M. Pavone _in

_1986,

when N =

_±03BC,

where _03BC =

min(x,y)~(0,0) |ax2+bxy+cy2l.

We

_only

need the

_special

case 03BC =1 of his result and

give

a self-contained

_proof,

_using

our unimodular matrix

_approach.

1. Introduction The standard

_approach

to

_solving

the

_equation

in

_relatively

_prime

_integers

x, y, is via reduction of

quadratic forms,

as in

Mathews

_([6,

_p

_97]).

There is a

_{parallel approach}

in Faisant’s book

([2,

_pp

106-113])

which uses continued fractions.

However,

in a memoir of

_1770,

_Lagrange

_([11,

Oeuvres

_II,

pp

655-726]),

gave a more direct method for

_solving

(l.l)

when

gcd(a,

b,

c)

=

gcd(a, N) =

1 and D =

b2 -

4ac _&#x3E; 0 is _{not a}

perfect

_square. This _paper seems to have

been

_largely

overlooked.

_(Admittedly,

the

_necessity

_part

of his

_proof

is

_long

and _{not easy} to

_follow.)

M. Pavone

_([10,

_p

_271])

solved

_(1.1)

when N = where

He had

_essentially

solved

_{( 1.1 )}

in

_general,

as

_Lagrange

showed how to

reduce the

_problem

to the case N = ~1.

(See (4.2)

and

(4.6)).

Strangely

Pavone made no mention of

Lagrange’s

_paper,

referring

instead

to Serret

_([12,

p

80]),

who had earlier drawn attention to the

possibility

of

an

exceptional

case.

A.

_Nitaj

has also discussed the

equation

in his

_thesis,

_([9,

_pp

_57-88]),

using

a standard

_convergent

sufficiency

condition of

Lagrange,

which

re-sulted in a restriction D &#x3E;

_16,

thus

_{making rigorous}

the

_necessity

_part

of

Lagrange’s

discussion.

_Nitaj

discussed

_only

the case b = 0 in

detail, along

the lines of Cornacchia

_(~l,

pp

66-70])~

Our contribution in this _paper is to use the

_convergent

criterion of Lemma

_2,

which results in no restriction on

_D,

while

_allowing

us to deal with the

_{non-convergent}

_case, without

_having

to

_appeal

to the _{case p =1} of

_Pavone,

whose

_proof

is somewhat

_complicated.

The continued fractions

_approach

also has the attraction that it

_produces

the solution

_{(x, y)}

with least

_{positive y}

from each

_class,

if

_gcd(a,

_N)

=1.

Our treatment

_generalises

an earlier _paper

_by

the author on the

_equation

x2 -

_Dy2

= N

(See

Matthews

[7]).

The

_assumption

that

_{gcd(a, N)}

= 1 involves no loss of

_generality.

For as

pointed

out

_by

Gauss in his

_{Disquisitiones}

_(see

_[3,

_p

_221]

_(also

see Lemma

2 of Hua

_[5,

_pp

_311-312]),

there exist

_relatively

_prime

_integers

such that

aa2

+

_ba,

+

cy2

=

A,

where

gcd (A, N) =

1. Then if a6 -

{3,

=

1,

the unimodular transformation x = aX ₊

#Y, y

=

~yX

₊ 6Y converts

ax2

+

_bxy

+

cy2

to

AX2

+ BXY +

CY2.

Also the two forms

_represent

the

same

_integers.

2. The structure of the solutions

We outline the structure of the

_integer

solutions of

_{( 1.1 )}

as

_given

in

Skolem

_([13,

_pp

_42-45]).

The

_primitive

solutions x +

of ax2

+

_bxy

+

cy2

= N

(i.e.

with

gcd(x,

y)

=1)

fall into

_{equivalence classes,}

with x +

yVD

and x’ +

being

equivalent

if and

_only

if

This is

_equivalent

to the

_equations

It is _{easy to}

_verify

that

_(2.1)

holds if and

only

if the

_following

_congruences hold:

Each

_primitive

solution

_gives

rise to a root n of the _congruence

In fact if

_{(a, 7)}

is a solution of

(1.1)

and a6 -

,~7

=

_1,

then

Equivalent

solutions

_give

rise to

_congruent

n

(mod

Conversely,

primitive

solutions which

_give

rise to

_{congruent n}

_(mod

are

equivalent.

This follows from the

_equations

and _congruences

_(2.3)

and

_(2.4).

It is also

_{straightforward}

to

_verify

that is

_{replaced by}

an

equivalent

form

AX2

+ BXY +

Cy2

under a unimodular

_{transformation,}

then

_equivalent

_{primitive representations}

_{(x, y)}

and

_{(x’, y’)}

of N _map into

equivalent primitive representations

(X, Y)

and

_{(X’, Y’).}

In fact the n

of

_equation

_(2.5)

is

_{replaced by An,}

where A is the determinant of the transformation.

_(See

Gauss

_[3,

_pp

_130-131].)

3. Some lemmas

Lemma 1. Assume D &#x3E; 0 is not a

perfect

_square and

Qo I (PO2 -

D).

the n-th

_complete

_convergent

in the

_simples

continued

fraction for x =

(Po

and =

QOAn-1 - POBn-1,

where

denotes a

_convergent

to x, then

or

equivalently

Lemma 2.

_{If w}

=

I

where

&#x3E; 1 and

P, Q, R, S

are

integers

such that

_Q

&#x3E;

_{0, S}

&#x3E; 0 and PS -

_QR

=

tl,

or S = 0 and

Q

= 1 =

R,

then

_{P/Q is}

a

_convergent

An/Bn

to w. Moreover

if Q ~

S &#x3E;

_0,

then

RIS

= _{+ +}

kBn), k

&#x3E; 0.

Also (+

k is the

(n

₊

l)-th

complete

convergent

to c~. Here k =

0 if Q

_&#x3E;

S,

while k &#x3E; 1

if

Q

S.

Proof.

This is an extension of Theorem

_172,

_Hardy

and

_Wright

([4,

_pp

140-141)),

who dealt with the case

Q

&#x3E; S &#x3E; 0. See Matthews

_[7,

_pp

325-326].

0

The

_following

result is a

_special

case

(p( f)

=

1)

of a result due to M.

Pavone,

[10,

p

271].

Pavone’s

proof

is rather

complicated

and we

_give

a

self-contained

_{proof using}

our Lemma 2 as cases

(ii)

and

(iii) (c)

of the

_proof

of Lemma 3 below and in the

_Appendix.

Lemma 3.

_Suppose

_X,

_y &#x3E;

_{0, Q,}

_n, R are

integers

and

where D

= n2 -

_4QR

&#x3E; 0 is not a

perfect

_square. Also let ca

= - 2~

and

w* _ be the roots

_of

_QB2

+ R = 0. Then either

(i) X/y

is a

_convergent

Ai-1IBi-1 to

w

(resp. w*)

and if

(Fi

+

denotes the i-th

_complete

_convergent

to w

(resp.

w*),

then

Qi

=

(-1)z2

(resp.

Q,

=

l /-l~i+12~~

_or

(ii)

D =

_5,

Q

0 and

where and

_As/Bs

denote

_{convergents to}

w and

_{w*, respectively}

and

where

Moreover

_{X/y is}

not a

_convergent

to c.r or cv*.

Conversely,

if

(i)

or

(ii)

_hold,

then

X/y

is a solution

of

(3.3).

Remarks.

_(a)

In the

_Appendix,

we _prove that if D = 5 and

Q

0,

then r - 1 - s

_(mod

₂₎

and

the latter

_being

obtained

_{directly by}

an

_appeal

to

_symmetry

_by

Pavone

([10,

p

277]).

The

_{corresponding equations}

hold if w is

_{replaced by w*,}

each

_Ar

is

_replaced

by As

and each

Br

is

_{replaced by}

_Bs

etc.

Consequently,

_A,-A,-,

is not a

_convergent

to w or w*.

Br-Br-i

(c)

If n is _{even, say n} =

2P,

then _w =

(-P

₊ and w* _

_{(-P -}

where A =

p2 -

QR.

If we then denote the

n-th

_complete

_convergent

to w

(resp. w*)

by

(P~

+ condition

_(i)

becomes

_Qi

=

(-1)z

(resp

Qi

=

(-1)z+1).

Proo f . Suppose

(3.3)

holds. Consider the matrix

where t =

_{-PX - Ry if n = 2P,}

Then in both cases,

Also it is

_{straightforward}

to

_verify

that

where

Case

_(i).

_{Suppose QX}

+

_Py

&#x3E; 0. Then as a &#x3E;

_1,

Lemma 2

_applies

and

X/y

is a

_convergent

to ar.

Case

_(ii).

_{Suppose QX}

+

_Py

= 0. On

substituting

for

QX

in

(3.4),

we

_get

Hence

Hence

Hence

_X/y

=

Lw* J

is a

_convergent

to c~* if

_{D ~}

5. ,

Then

Also

_QX2

+

_(2P

+

_1)X

+ R = 1 and P

= -QX

together

_give

Hence

- I - - -,

-and hence

_Q

0.

Case

_{(iii) (a)}

_{Suppose QX}

+

_Py

0.

_{Then -(QX}

+

_Py)

&#x3E; 0 and

where -a* &#x3E; 1 if

_{D ~}

5.

Hence

_X/y

is a

_convergent

to

_w*,

unless D = 5.

(b)

If D = 5

and -(QX

₊

Py) &#x3E;

_y, then

and

_again

_X/y

is a

_convergent

to w*

_by

Lemma 2.

In all cases where

X/y

is the

_convergent

to cv

(resp. w*),

it

follows from

_(3.3)

and

_equation

_(3.2)

of Lemma

_1,

with

_Po

= _-n,

Qo

=

2Q

(resp.

Po

=

n, Qo

=

-2Q),

that

and

_consequently

_(-l)’Q,,

= 2

(resp. -2)

in all _cases.

(c)

Now _suppose D = 5 and _y

&#x3E; -(QX

₊

Py)

_&#x3E; 0.

Now,

from

_(3.5),

Lemma 2 tells us that

Hence w* =

_{~ao, ... ,}

where s = i + 1

and as

= k _{+ 1.}

Hence (3.6)

gives

Next we _prove that

Q

0. We have from

_equation

_(1.1)),

However

so

and hence

Then

_equation

_(3.7)

_{gives Q}

0.

4. The main result Theorem 1.

_Suppose

where

_{N ~}

_0,

_{gcd(x, y) = 1 = gcd(a, N)}

_{and y}

&#x3E; 0 and D =

b2 -

4ac _&#x3E; 0

is not a

perfect

_square.

which is not a

_convergent

to w or w*. Also aN 0.

Conversely,

then

_(x,

_y),

with x =

yO

₊

INIX,

will be _a solution _to

(4.1),

_possibly

im-primitive.

Proof. Suppose

where

_{gcd (x, y)}

= 1 =

gcd(a, N)

and _{y &#x3E;} 0. Then

clearly

gcd(y, I N 1)

= 1. Let x -

_y8

_(mod

and

Then

Also

where

_Q

=

aiNI,

R =

(ao2

₊ b8 ₊

c)/INI

and

n2 -

_4QR

= D.

The conclusions of the theorem then follow from Lemma

_3,

_applied

to

the

_equation

+

_n’Xy

+

R’y2

=

_1,

where

Q’

=

EQ, n’

= _en,

R’

= eR

5. The

_algorithm

Let A =

D/4

if b is even and let the i-th

_complete

_convergent

to W or

W* be denoted

_by

+

-,/A-)/Qi

or

(Pi

according

as b is even or odd.

If

_equation

_(4.1)

is soluble with x =-

_y0

_(mod

&#x3E;

_0,

there will be

_infinitely

_many solutions because of

_equations

_(2.2).

It follows that if

w and w* are not

_{purely periodic,}

we need

_only

examine the first

period

the continued fractions for w and w* to determine

solubility

of

_(4.1).

_For,

with cv

(resp. w*)

being

(-P :f:

(resp.

(-(2P

+

₁₎

t

_VD)/2Q),

the

_equation

_{Qi =}

_(resp.

_f2)

will hold for

infinitely

many i

by periodicity

and so there will be at least one such i in

the range m i m + l.

_Any

such i must have

_Qi

= 1

(resp. 2),

_as

(Pi

+

01) /Qj

_{(resp. (P~}

+ is reduced for i _{in this range and} so

Qi

&#x3E; 0. Moreover if I is _even, the

_sign

of is

_preserved

from

one

_period

to the next. If I is

_odd,

then the first or second

period

will

produce

a solution. If w or w* is

purely periodic,

we must examine

_Q2,

which

_corresponds

to the third

_period.

Moreover there can be at most one i in a

_period

for which

_Qi

= 1

(resp.

2).

For if

_Pi

_{+ d1 (resp. (Pi}

+

VD)/2

is

_reduced,

then

_Pi

=

(resp.

Pi

=

2L(VD -1)/2J

₊

1)

and hence two such occurrences of

Qi

= 1

(resp.

2)

within a

period

would

_give

a smaller

period.

Hence we have the

_{following algorithm essentially}

due to

_Lagrange,

_apart

from

_stage

1:

1. If

_{gcd(a, N)}

&#x3E;

_1,

find a unimodular transformation of the

_given

quadratic

form into one in which

gcd(a, N)

= 1.

(See

the last

paragraph

of

the

_{Introduction.)}

2. _{Find all solutions 0 of the congruence}

_(4.3)

in the _range 0 0

(This

can be done as follows:

First

solve t2

=

b2 -

4ac

(mod

4INt), -INI

t

_(If

there are no

solutions

_t,

then there is no

_primitive

solution of

(4.1)

corresponding

to

t.)

Then solve a8 -

_t26

_(mod

_INI),

0 0

For each

_0,

let n = 2aO ₊

b,

P =

Ln/2j

and

Q

=

3. For each of the numbers c~ =

(resp.

-~2P2Q+‘~),

test the first

_period

to see if

Qi

= 1

(resp.

2)

occurs. If 1 _{is even,} test

_additionally

for 1 =

(-I)iN/INI

(resp.

2 = to hold.

Similarly

for each of the numbers w* -

-p- A

_(resp.

-~2

, with i

_{replaced by}

i + 1.

If D =

_5,

_test

additionally

_{to see} if aN 0 holds.

4. For each 0 and

_{corresponding}

for which test 3

_succeeds,

find the least i for which the condition

_Qi

=

(resp.

Qi

=

holds. If 1 is _even, this will occur in or before the first

period,

while if I is

odd,

this will occur in or before the second

period.

Similarly

for W*.

For the

_{corresponding}

_convergent

to w or

_w*,

write X =

y =

B$_1.

If D = _{5 and aN}

_0,

in relation _to

w, write X =

Ar -

Ar-l,

y =

Br - Br-i .

Then x =

y0

₊

produces

_a solution of

(4.1)

with x -

y8

(mod IND.

Choose the solution

_{(x, y)}

with lesser of

_{the y}

values.

The

_algorithm

will

_produce

a solution

(x, y)

from each

class,

with the

additional feature that the least

_{positive y}

is

_chosen,

if the

_quadratic

form satisfies

_{gcd (a, N)}

= 1.

6.

_Examples

Example

1

_(Gauss,

Article

_205).

_[3,

p

189]

As

_gcd

_{(42, 585)}

= 3 =

gcd (21, 585),

we make a suitable transformation

which

_gives

The latter form has A = 79 and

gcd(2, 585)

= 1.

We list the roots of

262

+ 180 + 1 - 0

(mod 585)

and

_{corresponding}

values P = 20 ₊ 9:

We find that

_only

P = 157 and 1013

_give

solutions of

equation

(6.1):

(i)

or =

(-157

₊

Jfi) /l170

gives Q3 = l,

A2/B2 = -1/7.

Then y’

= 7

and x’

= 7 - 74 - 585 ~ 1 = -67. Hence

(x, y) = (74, -141)

is a solution of

(6.1).

c~* =

(-157 -

79)/1170

also

gives

the solution

(74, -141).

(ii)

c~ =

(-1013

₊

79)/1170

gives

Q2

=

1,

AI/ B1

=

-6/7.

Then y’

= 7

and x’ = 7 - 502 - 585 6 = 4. Hence

(x,

y) = (3,1)

is a solution of

(6.1).

w* =

_{(-1013 -}

_v’79)/1170

also

_gives

the solution

_(3,1).

In fact Gauss gave solutions

(83, -87)

and

(3,1).

In the notation of

(2.2),

the solutions

_(x,

_{y) =}

_(83,

_-87)

and

_{(x’, y’)}

=

(74, -141)

are related

_by

the

solution

_{(u, v) = (-80, 9)}

of the Pell

_equation

_{x2 - 79y2}

= 1.

Summarising:

Example

2.

_{3x2 - 3xy - 2y2}

= 202. Here D = 33.

The solutions of

392 -

30 - 2 - 0

_{(mod 202)}

are

_{39, 63,140,1fi4,}

with

corresponding n

values

_{231, 375, 837, 981.}

Summarising:

There are 4

equivalence

classes of solutions.

Example

3.

=19x2 -

_85xy

+

95y2

= -671. Here D = 5.

The solutions of

1982 -

850 + 95 - 0

_(mod

₆₇₁₎

are

_{443,454,504,515,}

with

_{corresponding n}

values

_{16749,17167,19067,19485.}

The

_exceptional

solutions

_give

the solutions with smallest y:

(i) 8

= 443: w =

(-16749

₊

V5)/25498

=

[-1,2,1,10,1,1,2, I],

Exceptional

solution: . -- .... _

Exceptional

solution:

Exceptional

solution:

Exceptional

solution:

Summarising:

There are 4

equivalence

classes of solutions.

7.

_Appendix

Lemma 4. Let

Proof.

We have

and hence

Then

_(7.1)

_gives

and

_(7.2)

and

_(7.3)

_give

Hence

Also

_(7.2)

and

_(7.3)

_imply

Hence

It follows from cases

(ii)

and

(iii)(c)

in the

proof

of Lemma

_3,

that

and

Acknowledgement

The author is

_grateful

to John Robertson for

_improving

the

_presentation

of the paper.

The

_algorithm

has been

_implemented

in the author’s number

_theory

cal-culator program

CALC,

available at

_{http : //www . maths . uq.}

edu.

_{au/ ~krm/}

References

[1] G. CORNACCHIA, Su di un metodo per la risoluxione in numeri interi dell’ equazione

Chxn-h = P. Giornale di Matematiche di _Battaglini 46 _(1908), 33-90.

[2] A. FAISANT, L ’equation diophantienne du second _degré. Hermann, Paris, 1991.

[3] C. F. _{GAUSS, Disquisitiones} Arithmeticae. Yale University Press, New _Haven, 1966.

[4] G. H. HARDY, E. M. WRIGHT, An Introduction to _{Theory of} Numbers, Oxford _University

Press, 1962.

[5] L. K. _HuA, Introduction to Number _{Theory. Springer, Berlin,} 1982.

[6] G. B. _{MATHEWS, Theory of numbers,} 2nd ed. Chelsea _{Publishing Co.,} New York, 1961.

[7] K. R. _MATTHEWS, The Diophantine equation x2 - Dy2 = _{N, D &#x3E; 0. Exposition. Math. 18} (2000), 323-331.

[8] R. A. MOLLIN, Fundamental Number Theory with Applications. CRC Press, New York,

1998.

[9] A. NITAJ, Conséquences et _aspects expérimentaux des _conjectures abc et de _{Szpiro. Thèse,}

Caen, 1994.

[10] M. PAVONE, A Remark on a Theorem of Serret. J. Number Theory 23 _(1986), 268-278.

[11] J. A. SERRET _(Ed.), Oeuvres de _{Lagrange, I-XIV, Gauthiers-Villars, Paris,} 1877.

[12] J. A. SERRET, Cours _{d’algèbre supérieure,} Vol. _I, 4th ed. _{Gauthiers-Villars, Paris,} 1877.

[13] T. _{SKOLEM, Diophantische Gleichungen,} Chelsea _{Publishing Co.,} New York, 1950. Keith MATTHEWS

Department of Mathematics

University of _Queensland 4072 Brisbane, Australia