On the diophantine equation $x^2 = y^p + 2^k z^p$

S

AMIR

S

IKSEK

On the diophantine equation x

2

= y

p

+ 2

k

z

p

Journal de Théorie des Nombres de Bordeaux, tome 15, n

o

3 (2003),

p. 839-846.

<http://www.numdam.org/item?id=JTNB_2003__15_3_839_0>

© Université Bordeaux 1, 2003, tous droits réservés.

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

On the

_diophantine

equation x2

_{= yp + 2k zp}

par SAMIR SIKSEK

RÉSUMÉ. Nous étudions

_{l’équation}

du titre en utilisant une courbe de

_Frey,

le théorème de descente du niveau de Ribet et une

me-thode due a Darmon et Merel. Nous _pouvons determiner toutes les solutions entibres x, y, z,

premières

deux à

deux,

si p ~ 7

est

_premier

et k ~ 2. De

_cela,

nous déduisons des résultats

sur

quelques

cas de cette

_{équation qui}

ont ete étudiés dans la

litterature. En

_particulier,

nous _pouvons combiner notre résultat

avec les résultats

précédents

de Arif et Abu

Muriefah,

et avec ceux

de Cohn pour obtenir toutes les solutions de

l’équation

x2

+

2k =

yn pour n ~

3.

ABSTRACT. We attack the

_equation

of the title

_using

a

_Frey

curve, Ribet’s

_{level-lowering}

theorem and a method due to Darmon and Merel. We are able to determine all the solutions in

_pairwise

coprime integers

x, y, z if

_{p ~ 7}

is

_prime

and 2. From this we deduce some results about

_special

cases of this

_equation

that

have been studied in the literature. In

_particular,

we are able to combine our result with

_previous

results of Arif and Abu

Muriefah,

and those of Cohn to obtain a

_complete

solution for the

_equation

x2 + 2k = yn for n ~ 3.

1. Introduction

In

_[8]

Darmon and Merel

_study

the

_{diophantine equation}

x2

=

yp

+ z~

along

with two other variants of Fermat’s

_equation.

In this paper we show

that the method of Darmon and Merel can be

_adapted

to

_give

results about the more

_{general equation}

(1)

x2 =

yP

+

2kZp

where k is a

positive

_integer,

and _p a

prime.

Studying

this

equation

unifies

the

_study

of two well-known

_exponential

_{diophantine equations appearing}

in the literature:

x2+2 k

=yn and x2-2 k =yn

(see ~1~,

[3],

[5],

[6], [10]).

We shall call an

_integral

solution _{x, y, z} of

_equation

(1)

_primitive

if _{x, y,} z are

pairwise coprime,

and non-trivial if 0. The reader is forewarned

Manuscrit recu le 24 juin 2002.

This work is funded _by a _grant from Sultan Qaboos University, Oman

Theorem 1.

_Suppose

k &#x3E; 2 and that _p &#x3E; _{7 is}

_prime.

Then the

_{onl y}

non-trivial

_primitive

solutions

_of

_equation

₍₁₎

are k =

_3,

x =

~3,

_y = z =1 and

p

arbitrary.

In Section 3 we will

_give

the

_implications

of this theorem to the two

aforementioned

_{exponential equations.}

In

_particular,

we are able to

com-bine our theorem with earlier results of Arif and Abu

_Muriefah,

and of

J.H.E.

_Cohn,

to

_give

a

complete

solution to the

_{equation x2 + 2k}

=

Our

approach

in

_proving

Theorem 1 is to associate to each

_putative

solu-tion of

_equation

₍₁₎

a

_Frey

curve and

_apply

Ribet’s

_{level-lowering}

theorem to show that the Galois

_{representation}

on the

_p-torsion

is

isomorphic

to a

representation

of a much smaller level. This will be sufficient to eliminate all the cases of the theorem

_except

for k = 3 where we have to

_(slightly)

_adapt

the method of Darmon and Merel. Since we are

following

well-trodden

paths

here,

our

_exposition

will be rather terse.

_Apart

from

_[8]

the reader

may wish to compare what follows with

[7]

and

[12,

pages

397-399].

It has

_recently

come to my attention that W. Ivorra

[11]

proves

stronger

results for

_equations

₍₁₎

and

₍₇₎

_using

similar methods.

2. Proof of Theorem 1

Assume the existence of a non-trivial

_primitive

solution 2 and

p &#x3E; 7

prime.

We will show that this forces k = 3

and y

= z = 1. It is

helpful

to

_give

a

_complete

list of

_assumptions

we make about _{x, y, z,}

k,

_p

(1)

Equation

(1)

is satisfied.

(2)

k &#x3E; 2.

(3)

p &#x3E; 7 is

prime.

(4)

x, y, z are

_{pairwise coprime, odd,}

and non-zero.

(5)

x - 3

_{(mod 4).}

There is no loss of

generality

in

making

these

assumptions;

for the fourth

statement above we need to _{absorb any power} of 2

_{dividing z}

into the

2k

in

_equation

_(1).

For the fifth one we need to

_{replace x by -x}

if _necessary. 2.1. The

_Frey

Curve. Denote

_by

E the

_‘Frey

curve’

In standard notation we

have,

/A-2 ~~.~~3

Let

_Amin

be the minimal discriminant of

_E,

and N be its conductor.

Fi-nally,

if a is a non-zero

integer

then denote

_by

rad(a)

the

_product

of distinct

primes

dividing

a.

Lemma 1. The values

_of

_Amin

and N are

_{given by}

the

following

table:

&#x26; , .

Proof.

It is easy to show that c4 and A are not

_{simultaneously}

divisible

_by

any odd

prime.

We deduce thus that the curve is minimal and has

multi-plicative

reduction at all

_primes

_dividing

yz and at no other odd

_primes.

This shows that the table entries are correct modulo _powers of 2. It

re-mains to

_study

the reduction at 2. Recall that

x2

Thus we

may rewrite the

equation

for E as

Suppose

first that k &#x3E; 6.

_Replace

X

_by

4X - x and Y

_by

8Y + 4X. After

simplifying

we reach the model

I . 1 B

This model is

_integral

since we assumed that x = 3

(mod 4).

It is minimal at 2 since the ’new

_c4’

is

4x2 -

_3y~

which is odd. Hence this model is

global

minimal. The formulas for

_Amin

for the cases k = 6 and k _&#x3E; 6 _are

now immediate on

_noting

that the

_change

of variable above forces

Amin

=

2-12 Ll.

.

If k = 6 then

clearly

2 does not divide

Amin

and so it does not divide N. Thus _suppose k &#x3E; 7. The reduction modulo 2 of the model

₍₄₎

is

In either case the

_singularity

at

_(0,0)

is a node. This shows that 2 divides

the conductor ~V

_precisely

once.

For k =

2,3,4,5,

it is clear that the curve is

_already

minimal at 2 since

212

does not divide A. The rest now follows from Tate’s

algorithm.

For

the benefit of the reader who would like to

_verify

the details we

_point

out

that it is best to make the

_following

_changes

of variable in the model

₍₃₎

2.2. The Galois

_{Representation.}

Now let

be the Galois

_{representation}

on the

p-torsion

of E. To be able to

apply

Ribet’s

_{level-lowering theorem,}

and

_complete

our

proof,

we will need to

show that _p is irreducible. For this we need two

_lemmas,

the first of which is

_elementary.

Lemma 2. The

_{equation a2}

= :1:1 :I: 2~ has _no solutions

for

m &#x3E; 2

_except

for32=1+23.

_

Lemma 3. The

_{j-invariant of}

E is

_{non-integral except i f}

k = 3 and _y = z=1.

Proof. Suppose

that the

_j-invaxiant

is

_integral.

It is _easy to use this and the

_assumptions

listed at the outset

of the

_proof

about

_{x, y, x, k, p}

to show

_{that y}

== ±1 and z = ±1. From

equation

(1)

we see that

x2

= +1 ±

2k,

and this forces k = 3

and y

= z = 1

by

the

_previous

Lemma. 0

Lemma _{4. p is} irreducible.

Proof.

If we are in the case k = 3

and y

_{= z} = 1 then the _curve E is

curve 32A4 in Cremona’s tables

[4]

and so we know that it does not have

isogenies

of odd

_degree.

Thus _p is irreducible.

_Suppose

we are not in this

case. We note the

_following,

~ E has a

_point

of order 2. This is clear from the model

(2)

_given

for

E. ’

~ The

_j-invariant

of E is

_non-integral

(by

the

_previous

lemma).

~ E has some odd

_prime

of

multiplicative

reduction.

Otherwise,

from

the formulas for the conductor

_given

in Lemma 1 we see

that y

=

and z

_{= ::1:1,}

which is

_impossible

as above.

These three

_properties

are

enough

to force the

irreducibility

of _p for

p &#x3E; 7;

for this see the

proof

of

[8,

Theorem

7].

0

2.3. Conclusion of the Proof. It has been shown that all

_elliptic

curves

over Q

are modular

[2].

However,

we note in

_passing

that our

_Frey

curve

E has semistable reduction _away from 3 and 5

_(by

Lemma

_1),

and so

its

_modularity

follows from an earlier theorem of Diamond

[9]

which is a

Let

_N(p)

be the Serre conductor of p

(see

[15,

page

180]

for the

def-inition).

It follows from the

_properties

of the Serre conductor and our

Lemma 1 that

For this see

[15,

_page

207]

or

[7,

_page

264].

Applying

Ribet’s

level-lowering

theorem

_[14]

we see that _p

_corresponds

to a

_{mod p}

eigenform

of

weight

2

and level

_2,4,8

or 16 if

k ~

3 and of level 32 if k = 3. Since there are no

weight

2 cusp forms of levels

2, 4, 8,16

we

immediately

deduce that k = 3.

We will henceforth assume that k =

3,

and _to

complete

_our

proof

_we

must show that _y = z = 1. The rest of our

proof

will mimic the

approach

made

_by

Darmon and Merel

_[8,

_pages

_88-99]

for the

_equation

xP

_{+ yP}

=

z2

(which

really corresponds

to the case k = 0 of our

_equation

1).

We saw

above that _p

_corresponds

to a mod _p

eigenform

of

weight

2 and level

32,

and we know that our

_Frey

curve E _possesses a

_point

of order 2. This is

_exactly

the same situation as in Darmon-Merel for their

_equation

mentioned above.

Their

_arguments

show that if p - 1

_{(mod 4)}

then E has

_potentially

_good

reduction at all odd

_primes

_(see

the

_proof

of

_[8,

_Proposition

_4.2]).

As before this is

_{just saying that y}

= tl and z = ~1. We

immediately

_see

that y

= 1

and z = 1

(since

k =

3).

We

may thus assume that _{p -} 3

(mod 4).

Proposition

4.3 of

_[8]

shows that the

_image

of _p is

_isomorphic

to the normalizer of the

_non-split

Cartan

_subgroup.

Then Theorem 8.1 of

_[8]

shows that the

_j-invariant

of E

_belongs

to

_Z[~].

_{Moreover, p}

does not

divide _yz

_by

the same

_argument

as in the

_proof

of

_[8,

_Corollary

_4.4].

Hence

we know

that j

is in

Z,

and from Lemma 3 above we know

_{that y}

= z = 1

as

required.

2.4.

_Exceptions

to Theorem

_1;

the cases _p 7 and k = 1. The

reader will

_probably

be

_wondering

where our

_proof

fails for the _{cases p} 7

and k = 1.

For p

7 _{we are} unable _to

_prove

the

irreducibility

of the Galois

representation

p. For k = 1 the Serre conductor

of p

is

_128,

and _we know

that the dimension of

_SZ(ro(128))

is 9. So far as we

_know,

there are as

of

_yet

no methods

_developed

to deal with cases where the dimension after

level-lowering

is

_greater

than 1.

3.

_Special

Cases

3.1. The

_{equation x2}

+2 k

=

yn.

In

[5]

J.H.E. Cohn

shows,

for k

odd,

that the

_equation

for a &#x3E; 0.

The case with k even has

proved

to be more troublesome. In

[1]

Arif

and Abu Muriefah made a

_{plausible conjecture}

as to the solutions with k = 2m. Cohn

[6]

proves the

conjecture

for most values of m less than

1000, including

all those less than 100.

_Equation

₍₅₎

is of course a

_special

case of

(1)

corresponding

to the case z = -1

provided

that n =

p is an

odd

_prime.

It turns out that our Theorem 1 is

_enough

to

_complete

the resolution of

_(5).

Theorem 2.

_(Conjecture

_of

_Arif

and Abu

_Muriefah)

_{If n &#x3E;}

3, k

=

_2m,

the

_{diophdntine equations}

₍₅₎

has

_precisely

two

_{families of}

solutions

_given

for

and

11-23M

3M + 1.

Proof.

We shall use the results of

~1~, [6]

to reduce the theorem to a

special

case of Theorem 1. It is shown in

_[1,

_page

_300]

that n must be

_odd,

and that there are no further solutions with _{x, y} even. It is

_enough

therefore to

show that there are no solutions to the

_equation

for m &#x3E;

_0,

_p &#x3E; 3

_prime,

and _{x, y} odd

_integers.

However Cohn showed that

any such solution to

equation

(6)

must

satisfy

m &#x3E; 100

([6,

_page

462])

and

p =

_{1, 4, 7}

_{(mod 9) ([6,}

Lemma

_3]).

In

_particular,

_any such solution will be

a solution to

_equation

₍₁₎

with _p &#x3E;

_{7, z}

= -1 and k = 2m &#x3E; 200. This

contradicts Theorem 1. 0

Remark. M. Le has

_{just published}

a solution to

_equation

₍₆₎

_using

linear forms in

_logarithms

_{(see [13]).}

3.2. The

_{equation x2 -}

2k

=

yn.

The

equation

has been considered

_by

_Bugeaud

_[3]

and

_by

Guo and Le

_[10].

In

_particular,

Bugeaud

showed that

_{if x,}

_y,

_{k, n}

is a solution

_satisfying

the above conditions

then,

o n and k are

odd;

this was shown

_{using elementary}

_{factorization}

ar-guments.

o n 7.3 x

105;

this was established

_using

lower bounds for linear forms

in

_logarithms.

While we are unable to resolve

_equation

₍₇₎

_completely,

our Theorem 1

Theorem 3.

_If

the

_equations

₍₇₎

has a solution

satisfying

the above

condi-tions then either k = 1 _{or n} =

3a5b for

_{some a,} b.

We note in

_passing

that it

_maybe

_possible

to eliminate the case n =

3a5b

from the theorem as follows.

Clearly

it is sufficient to solve the

equations

Now k must be odd

_by

the result

_quoted

_above,

and this leads us to fac-torize the left-hand side of each

_equation

over We can reduce each

equation

to a set of Thue-Mahler

_equations,

and these can be solved

using

standard methods as in

[16,

Chapter

VIII],

although

we have not

attempted

to do this. We do not see how the

_{equation x2 -}

2 =

yn

(corresponding

_to

k =

1)

_can be solved

using

existing

methods.

Acknowledgment.

I am

deeply

indebted to the referee and Professor

_Bjorn

Poonen for many useful comments and

corrections,

and

particularly

for the

suggestion

that _my method can deal with

_equation

(1)

and not

_just

the

equation

(5).

References

[1] S. A. ARIF, F. S. ABU MURIEFAH, On the _{diophantine equation x2} + 2k = _yn. Internat.

J. Math. &#x26; Math. Sci. 20 no. 2 _(1997), 299-304.

[2] C. _BREUIL, B. _CONRAD, F. _DIAMOND, R. _TAYLOR, On the _{modularity of elliptic} curves

over Q: wild 3-adic exercises. J. Amer. Math. Soc. 14 (2001), 843-939.

[3] Y. BUGEAUD, On the _diophantine

_{equation x2 -}

2m = _±yn. Proc. Amer. Math. Soc.

125 _(1997), 3203-3208.

[4] J.E. _{CREMONA, Algorithms for} modular _elliptic curves (second edition). _Cambridge

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203B2yp

= z2 et xP +

203B2 yp

= 2z2. _{To appear in Acta}

Arith.

[12] A. W. _{KNAPP, Elliptic} curves. Mathematical Notes 40, Princeton _{University Press,} 1992.

[13] M. _LE, On Cohn’s _{conjecture concerning} the Diophantine equation x2 + 2m = _{yn, Arch.}

Math. 78 no. 1 _(2002), 26-35.

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[18] A. WILES, Modular _elliptic curves and Fermat’s Last Theorem. Ann. Math. 141 _(1995),

443-551. Samir SIKSEK

Department of Mathematics and Statistics

College of Science

Sultan _{Qaboos University}

P.O. Box 36

Al-Khod 123, Oman