Minimal foliations on lie groups

Indag.

Mathem.,

Minimal

N.S.,

foliations

by Enrique

3 (l), 41-46

March

30, 1992

on Lie groups

Macias-Virgbs’

and Esperanza

Sanmartin-Carb6n’

’ Departamento de Xeometria e Topoloxia, Universidade de Santiago de Compostela, Faculdade de Matema’ticas, E-15706Santiago, Spain 2 Departamento de Matemdticas, Universidade de Vigo, E-36271- Vigo, Spain

Communicated

by Prof.

W.T.

van Est at the meeting

of June

17, 1991

ABSTRACT Let F(G,H) group

be the foliation

G. We prove

F(G, H) are minimal

determined

that there always

by a (not necessarily

exists a Riemannian

closed)

metric

Lie subgroup

on G for which

H of a Lie the leaves of

submanifolds.

1. INTRODUCTION

Let F(G,H) be the riemannian foliation determined by an analytic Lie subgroup H of a connected Lie group G. The aim of this paper is to prove the existence of some (bundle-like) riemannian metric on G for which the leaves of F(G, H) become minimal submanifolds. In [TY] examples are given of foliations F(G,H) which are minimal but not totally geodesic, for left invariant metrics under rather restrictive hypotheses, particularly that H admits a bi-invariant metric. We give a direct proof for F(G, H) of Molino of Fedida’s theorems about the structure of transversely complete foliations. This enables us to restrict the problem to abelian Lie foliations F(K,N) with N a dense connected Lie subgroup of its closure K. In this case minimality holds for some (any, if H is unimodular) left invariant metric on K. For arbitrary foliations F(G,H), the desired metric is constructed by glueing together the fibres of G + G/K. Finally, we give an example where there is no left invariant minimizing metric. 2. PRELIMINARIES

2.1.

Let (M, F) be a foliated manifold.

The foliation

Fis said to be Riemannian 41

if there exists a Riemannian metric g on It4 which is bundle-like for F, that is the functions g(Y,Z) are constant along the leaves for all foliate, orthogonal to the leaves, vector fields Y Z defined on an open set of M. Recall that foliate , vector fields Y are defined by the condition [Y TF] C TF. We say that F is minimal for a given riemannian metric on A4 if the leaves are minimal submanifolds. This means that the Weingarten’s maps WY : TF+ TF have zero trace for all normal vectors Y. A particular case is that of totally geodesic foliations, where WY= 0, i.e. leaves are totally geodesic submanifolds. 2.2. Let G be a connected Lie group, Han analytic subgroup of G, that is H is a connected Lie group and there is an injective morphism of Lie groups from H into G. Let F= F(G, H) be the foliation on G whose leaves are the left cosets gH, g E G. Takagi and Yorozu proved the following: THEOREM [TY]. Ad(H)-invariant geodesic for g.

Let g be a left invariant riemannian metric on G. If g is then g is bundle-like for F(G,H) and F(G,H) is totally

We first observe that in this case the restriction gK of g to the closure K of H is a bi-invariant metric on K. In fact, g is Ad(K)-invariant because H is dense in K and Ad : G+ CL(g) is a continuous map. As a consequence, both K and H are unimodular Lie groups. Let g (respectively h) be the Lie algebra of G (resp. H). We shall use the following rank of indexes: 1 5 i,j, k, . . . I m;

lra,fl,x

,... In,

if the dimension of the group is m + n and the dimension of the subgroup is m. PROPOSITION 1. F(G, H) is minimal for a left invariant metric g on G if and only if Cr=, ci, = 0, where {c,“,} are the structure constants relative to some adapted to b g-orthonormal basis of g.

The Weingarten’s operator associated to a normal vector field Y defined in a neighborhood of XE G is given by PROOF.

W,‘(X) := -prx(VXY),

XE T,F(G, H)

where pr,: T,G -+ T,F(G, H) is orthogonal projection. By the very definition, F(G, H) is minimal if and only if trace W,‘=O for a11 normal vectors Y, for all XEG. When g is a left invariant metric on G, it suffices to prove the nullity of the trace at the identity of G. Let {X,, . . . . X,, Y,, *a*,Y,,} be any adapted orthonormal basis of g. We have trace WY” =,t, g(W’“(Xi),Xi) = -,;, g(K, 42

= -i g(Vx, Ya,Xi) i=l

Y,l,4) = -; ci,. r=L

0

3. THE STRUCTURE

The closure

OFF(G,H)

of N in G is a connected

closed

Lie subgroup

K of G, and

rc : G -+ G/K is a locally trivial (principal) bundle whose fibres are the closures of the leaves of F(G,H). Let us call n the basic fibration of F(G, H) and

W= G/K the basic manifold. The restriction of F(G, H) to each fibre defines a foliation isomorphic to F(K,H). We make use of the following result from Mal’cev:

of n

Let H be a dense connected Lie subgroup of a Lie group K. Then H is normal in K and the quotient f/t) of their Lie algebras is an abelian Lie algebra.

THEOREM

[Ma].

F(K,H)

Thus

is an abelian Lie foliation

with dense leaves as defined

by

Fedida [F]. As a matter of fact F(G, H) is a transversely complete foliation in the Molino’s sense [MO]. We now give for this particular case a version of the Fedida’s structure theorem. Let R be the universal covering of K. Let f: R+ I?“, n = codim F(K, H), be the morphism of simply connected Lie groups associated to the morphism E-+ f/l~ of Lie algebras. Then the closed subgroup A= ker f of I? is just the connected Lie subgroup of R with Lie algebra 6. Moreover I? is the universal covering of H. This means that the lifting of F(K, H) to R is the foliation F(J?, I?), which is simple for the fibration f: R-, R”. We remark that this construction makes it possible to easily obtain a bundlelike metric for F(G, H), starting with any metric go (that one can suppose it is left invariant) on G, the usual metric on IF?“, and any metric g, on the basic manifold. Takagi and Yorozu’s hypotheses imply that g, can be taken as induced by gG.

4. MINIMALITY

OF

F(K,H)

We first study minimality of the foliation F(K, H). That is, His a dense connected Lie subgroup of a Lie group K. When K is compact, the foliation is minimal by a result of Haefliger on abelian Lie foliations [Ha]; the theorem of Takagi and Yorozu cited above also applies because there is a bi-invariant metric

on K. Our next result includes

the non-unimodular

case.

2. Let H be a dense connected Lie subgroup of a Lie group K. Then there exists a left invariant riemannian metric on K such that the foliation F(K, H) is minimal. THEOREM

We shall divide the proof

into several

PROPOSITION 3. If H is not unimodular, left invariant metric.

propositions.

then F(K, H) is minimal for some

43

Let {Xi, . . . ,X,, Yi, . . . , Y,} be any basis of the Lie algebra f of K, adapted to the Lie subalgebra IJ of H. Let us consider a change of basis PROOF.

x;=xi,

i=l,...,

m.

Yi = i C7hXj+ Y,,

a= 1, . . ..n.

j=l

Let (c,“,} be the structure constants relative to the first basis. Since f/Ij is an abelian Lie algebra, the structure constants {CA%}of the new adapted basis verify c,$ = c$; c,z= C ahe:+&; ‘k CaB=

i c$ + C aha+:

+ C a:$-

C a&.

i

i,i

i

Let g be a left invariant metric so that {Xi, . . . . X,!,,,Y,‘,. . . . Y,l> is an orthonormal basis of E. Then F(K,H) is minimal for g if and only if (Prop. 1) 0= f ciA= i a,k(fJ c/i)+ fj cji i=l

k=l

j=l

for all a,

J=1

that is (1)

C(ak ... a,“)‘= -i j=l

ci Ja’

a=1

9 *.a,

n,

Since H is not unimodular, there exists i such that cy=, CL# 0. Then rank C= 1 and system (1) can be solved for all (r. 0 We remark that each solution of system (1) in the preceding proof gives a different distribution complementary to the foliation. Then, there is an infinity of left invariant minimizing metrics with different orthogonal distributions. For H an unimodular group, we shall need the following: Let H be a dense connected Lie subgroup of a Lie group K. If H is unimodular then K is unimodular.

PROPOSITION 4.

We must prove that det Ad&) = 1 for all y E K. As we have seen at $3, F(K, H) lifts up to the simple foliation F(x,fi). If n : e---t G is a covering, then Ad&x) =Ado(n(x)) for all XE e. This shows that fl is unimodular. Now, A is a normal closed subgroup of R, then det AdK(h) = det Ad&(h) for all h EA because AdR,&h) =Id. In fact, in our case Ad,,, is trivial because VI) is abelian. Thus det Ad,(h) = det Ad,(h) = 1 for all h E H. Finally, let y E K. Since H is dense in K there exists {h,} C K with y = lim h,, then det Ad&) = lim det Ad,(h,) = lim det Ad&h,) = 1. Cl PROOF.

44

5. Let H be a connected dense Lie subgroup of a Lie group K. If H is unimodular then F(K, H) is minimal for any left invariant metric on K.

PROPOSITION

Let g be any left invariant metric, (X,, . . . ,X,, Yr, . . . , Y,} an adapted PROOF. to h orthonormal basis of f with structure constants {c,&}. Since K and H are unimodular and f/h is abelian, we have 0 = trace adp (Y,) = 5 Cij J=I

which is equivalent to minimality by Prop. 1. 5. MINIMALlTY

0

OF F(G,H)

We now study minimality of the foliation F(G, H), with G a connected Lie group and H an arbitrary analytic subgroup. 6. Let (Iv&F) be a foliated manifold such that the closures of the leaves are the fibres of a locally trivial bundle p : M-t B, with fibre N and structure group K. Let g, be a riemannian metric on N such that the induced foliation on the fibres is minimal. If K acts by isometries on the fibres, then there exists a riemannian metric on M such that the foliation F is minimal. THEOREM

Our proof is inspired by an argument of Ghys [G]. Let { Uj} be a covering of B by coordinate open sets. Let us denote B; :p~‘(Ui) --f Ui XN the trivialising maps, @i=pro 0i with pr: Q xN+ N the projection. On Q x N we consider the metric gi = ( >ix g,, where ( )i is the pull-back I+v:<> of the usual inner product in R” by a local chart (y, I,v;) on B. Let {A} be a partition of unity subordinated to { Ui}. We define on A4 the metric g = C (A op) O,*g,. We shall verify that F is minimal for g. Let x~A4, Y, a vector normal to F at x, T/,E T,F a vector tangent to F. Then PROOF.

2g,(P,

r, v = Y&v, v

+2&w

iv,Yl)

= C Y,<& oP)(e,*g;),(K

v, + C (A oP)(x)y~(e:gi(Kv)) I

I2 c f(P(x))(qg;),([l/

Yl, V)

+ C .&(PCx)>{ Yx(e*gj(K v)) + 2(e*gi),([V y19 v)}

+ C

J(P(x))2(e,Xgi),(~1

Y,V 45

where V’ is the Levi-Civid Vj$p,Y+ V&&* Y, then

connection

associated

to O,+gj. But I~~,V,$Y=

(e,*g;),CvtX v, = g;Q,xj(v~*~;P*xy+ v~~~VOi*x~ei*xv)

Finally, we have 2g,(V,Y, U = c r,(& ~P)gNm,(,)(~i*, v, @;**V) I

+ 2 c A (P(X)) gN@+, (v~exV4;*x~@i*xv) I

where V N denotes the Levi-Civita connection of g,. Let { wjj : q.n Uj- K) be the transition functions of the bundle, @J;(X) = ~,~(p(x))@~(x). So Qi(x) and @j(~) differ in a translation on the fibre p-‘(p(x)) by an element of K. But K acts on the fibres by isometries, then

&(VV,Y,V = c x(P(x))gN~,,,(V~*~~~i*,Y,~~*~~) I

because Y,(c, (A 0~)) = Y,(l) = 0. basis of T’F, then trace Let {Vi,..., V,} be an orthonormal -C,“=I gx(VQ x vj) = 0 because (N, F) is minimal for gN. 0 In our case, B = G/K, the structure group K acts by left translations, metric minimizing F(K,H) is left invariant. We have then proved:

Wxy=

and the

I. Let G be a connected Lie group, H a connected, not necessarily closed, Lie subgroup of G. Then the foliation F(G, H) is minimal for some riemannian metric on G. THEOREM

The following example was pointed out to us by A. Reventos, showing that in general it is not possible to find left invariant minimizing metrics. Let g = (x, y, z> be a Lie algebra given by [x, y] = 0, [x, z] =x, [y, z] E (x, y) arbitrary. Consider fi = (x). Then for an adapted basis (Ax, Y Z), Proposition 1 would imply [Ax, Y] = [Ax,Z] = 0, hence Y,Z E (x, y). REFERENCES [F]

Fedida,

[G]

Ghys, E. - Feuilletages riemanniens sur les varietes simplement Grenoble 34(4), 203-223 (1984).

E. - Feuilletages

de Lie. These Strasbourg

[H]

Haefliger,

A. - Some remarks

on foliations

(1983).

with minimal

connexes.

leaves. Journal

Ann. Inst. Fourier, of Diff. Geom.

15,

269-284 (1980). [Ma] Mal’cev, A. - On the simple connectedness of invariant subgroups of the Lie groups. Acad. Sci. U.R.S.S. 34, JO-13 (1942). [Mo] Molino, P. - Riemannian Foliations. Progress in Math., Vol. 73, Birkaiiser (1988). [TY] Takagi, R. and Yorozu, (1984).

46

S. - Minimal

foliations

on Lie groups.

Tohoku

Math.

J. 36, 541-554

Indag.

Mathem.,

Exponential

N.S.,

3 (l), 47-57

diophantine

March

equations

30, 1992

with four terms

by MO Deze and R. Tijdeman Department of Mathematics, Leizhou Teacher’s College, Zhanjiang, Guangdong, P.R. China Mathematical Institute, R.U. Leiden, P.O. Box 9512, 2300 RA Leiden, the Netherlands

Communicated

at the meeting

of December

16, 1991

ABSTRACT By Theorems

1, 2 and 3 it becomes

a simple matter

to solve any equation

pX-qY=n, ~qY+pZ*qwf

1=0

x, y, z, w, where p and q are distinct

in non-negative

integers

positive

with ns5000.

integer

or pxkqy*pz*qw=O primes

below

200 and n is some

INTRODUCTION

1.

In 1983, P. Vojta

[6] stated

the following

result.

Let S be a finite set of places of Z containing at most three elements. Fix integers a, b, c, d. Then there are only finitely many solutions to the equation THEOREM

(1.1)

(Vojta).

aX+bY+cZ+d=O

in S-units X, Y, Z; and these solutions can be effectively bounded in terms of a, b, c, d and S. The theorem is only true under the additional hypothesis that the left side of (1.1) has no vanishing subsums. We call such solutions non-degenerate. The ef-

47

fective bounds

follow from known

lower bounds

for complex

and p-adic

linear

forms in logarithms of algebraic numbers. Let p, q be distinct prime numbers and assume that (1.1) is satisfied by rational numbers X, Y, 2, whose denominators are composed of p and q. By multiplying the lowest common denominator (and dividing by the greatest common divisor) (1.2) in

we obtain

a~+bI’+&+d~=O

integers

2,

Y,

gcd(aX, b7, &, d@) = b, c, d is divisible by p so pfz”@say. Further have essentially a three (1.3) If q{xP, (1.4)

an equation

2,

@

composed

of

the

primes

p,

q

subject

to

1. Without loss of generality we assume that none of a, or q. It follows that at most two terms are divisible by p, at most two terms are divisible by q. If q{.f@, then we term equation. If q’/ I?‘, then we arrive at an equation

apXqY+bpZ+cqW+d=O

(X,Y,Z, WE&o).

then we arrive at an equation apx+bpy+cq”+dqw=O

(-%Y,Z, WE&l).

All other cases can be transformed into (1.3) or (1.4) by a suitable permutation. Skinner [3] has explained how estimates for linear forms can be applied to solve equations (1.3) for given p, q, a, b, c, d in practice. Some authors studied the special case /al = JbJ = JcJ = JdJ = 1 of (1.3) and (1.4). Brenner and Foster [I] solved numerous four and even five terms equations, not necessarily with bases composed of only two primes. They used congruences and remarked (Comments 8.033 and 8.037) that the classes of equations (pq)x-pz - qw + 1 = 0 and px-py+ qz- q”‘=O do not seem to be amenable by their methods. In 1988 Tijdeman and Wang [5] solved all cases of the equations (1.5)

pxqy’pz+q”*

1 =o

(x,y,z,wE~>cl)

and (1.6)

pX+pYfqZ+qW=O

(x,Y,z,~E~>O)

for p=2, q= 3 with two positive and two negative terms. In particular, their result shows that in Comment 8.031 of [I] the solution (3,1,1,2) is missing and that there are no more solutions (see [5] Lemma 5). In 1989 Wang [7] treated all cases of (1.5) and (1.6) for p = 2, q = 3 with three positive and one negative term. In this paper we extend the latter results. Let p and q be prime numbers with p< q< 200. In Theorem 1 we show that 2 I5 is an upper bound for all powers in the non-degenerate solutions to the equations (1.5). In Theorem 2 we show that 215 is also an upper bound for all terms of non-degenerate solutions to the equations (1.6). It is now easy to determine the complete set of solutions for each single equation, as we illustrate in Section 5. As an easy consequence of Lemma 1 we show in Theorem 3 that if p” and qy differ by no more than 5000, then ~~129~. We give the six solutions with 48

pX~200’ explicitly. This result can be compared with a result of Styer [4] who computed an upper bound for the equation apX- cqy = n with p and q distinct primes less than 14 and a, c and n positive integers with as50, ~~50 and nllOO0. Styer does not apply estimates obtained by Baker’s method. Our method is based on a result of de Weger, which is proved by using estimates obtained by Baker’s theory of linear forms in logarithms. This method is applicable for any pair of primes p, q so that (1.5) and (1.6) can be solved for any prime pair p, q. In a similar way Theorem 3 can be extended to any prime pair p, q and any positive integer n. The results in this paper are based on the homogeneous inequality (3.1). In [9] de Weger has worked out how to solve the corresponding inhomogeneous inequality

for given integers e, f, primes p, q and real BE (41). Using such a result it is possible to solve equations (1.3) and (1.4) for arbitrary a, b, c, d (cf. Skinner [3]) and Styer’s equation ap” - cqy = n for any given integers a, c, n and primes p, q. We thank dr. B.M.M. de Weger for some valuable suggestions. 2. RESULTS

Let p and q be distinct primes less than 200. The first result deals with the equation (2.1)

pxqY+pZ+qW*

1 =o

in non-negative integers x, y, z, w. The case zw = 0 is obvious. The numbers c~,~ are defined at the bottom of the next page. THEOREM

1.

Every solution of (2.1) with zw> 0 satisfies

max($,qy,pZ,qW)1215. Additionally,

min(px,pZ)rcP,,w,

min(qY,qW)Ic,,,z,

pxqyIpz+qw+

1.

The second theorem deals with the equation (2.2)

py’qY+pZ+qW=O

in non-negative integers x, y, z, w. Because of symmetry it is no restriction to assume x2 z, yz w, px> qy. The case xy = zw is easy. THEOREM

2.

Every solution of (2.2) subject to XLZ, yr w, pX>qy, xy>zw

satisfies max(p”, qy, p”, qW) 5 215. REMARK.

If pX=2”,

qy= 181*, then we have the following solution of (2.2)

215- 181*-2’+ The latter equation

181”=0.

can also be written

as a solution

of (2.1), namely 49

215. 181°-23-1812+1=0. theorems THEOREM

(2.3)

This

shows

that

the

upper

bound

215 in both

is the best possible. 3.

Let n be a positive integer with n I 5000. AN solutions of

$--qy=n

satisfy max(p”, qy) 4 294 = 707281. The only solutions with px> 1992 are given by 294-893=2312, 433-57= 3. AUXILIARY

58-733=1608,

2’7-194=751,

174-433=4014,

1382, 413-216=3385.

RESULTS

In the proofs of Theorems 1 and 2 we use Lemmas l-4. Lemma 1 is based on a result of de Weger, Lemmas 2 and 3 are straightforward and Lemma 4 is a refinement of a result of Petho and de Weger [2]. Theorem 3 is a direct consequence of Lemma 1. 1. Let p and q be distinct primes less than 200. The non-negative integer solutions k, 1 of the diophantine inequalities

LEMMA

(3.1)

0 < pk - q’ < (pk)0.‘6

satisfy pk
According tine inequalities

to [8] Theorem

4.3(a) (= [9] Theorem

5.2(a)) the diophan-

0
and q’> 1015, then

whence

On checking all pairs pk, q’ with q’ 1992, we find exactly 27 pairs satisfying (3.1). These pairs lead to the exceptional values for p in Table A. For all other pairs satisfying (3.1) we therefore have pk< 1992
50

2.

Let p and q be distinct primes and u E iZ,o. Then

ordp(q* + 1)s

log(cp,qu) logp

PROOF.

Clearly,

for every

.

positive

integer

h, phv is the smallest

positive

integer s such that qs+ 1 is divisible by~“‘~ but not by P~+~+‘. It is therefore 0 sufficient to prove the lemma for u = v. This is obvious. LEMMA

(3.2)

3.

In the notation of Lemmas

cp,4< log 4 -

b) (3.3)

ACP logq

4

1 and 2 we have for p< 200, q< 200

187.49 if (p,q)=(137,19) 113.24 if (p,q)=(29,41) 98.75 if (p, q) = (97,53) 78.50 if (p,q)=(101,181) 60.60 otherwise;

I %

+ -L-

P

I 188

for all pairs p, q;

logp

c) 8207 if (p,q)=(lOl,

181) or (181,101)

(3.4) d) if (p, q) = (2,127) if (p, q) = (3,163) if (p,q)=(7,19) if 25~17 and (p,q)#(2,127), (3,163), if (p,q)=(29,41) p2-1 if 171~131 and (p,q)#(29,41) 3527 if (p,q)=(137,19), (97,53) or (101,181) -1140 in all other cases. 602 300 731 230 2130

(3.5)

PROOF. LEMMA

By straightforward

4. X5

PROOF.

Let az0,

calculation

of c~,~ and use of Table A for p.

b>O, x>O. Zf xsa+blogx,

(7,19)

0

then

&a+blogb).

Put 6 = e/(e - 1). If b log x> (6 - 1)a + 66 log b, then a<&log($),

51

whence xla+blogx<6(6-1))‘blog(x/b). 6(6 - 1))’ log y, which is a contradiction.

Put x-by. Then we infer y< Thus x~a + b log XI 6(a + b log 6). 0

4. PROOFS OF THE THEOREMS OF THEOREM 3. We first show that log max($, qy) 5 max(j3,11.25) where j? is given in Table A. If max(p”, qy)>eP, then we have, by Lemma 1,

PROOF

lp” - qy / 1 max($, qy)o.76. Hence

which shows our claim. On checking all pairs p”, qy with qy
z>o,

OF THEOREM

1.

Suppose that (2.1) holds for integers x20,

w>o.

CASE I. xsz

and yew. We have, by Lemma 2, (4.1)

q’s

Px=p,4w,

cq#z*

We first show that (4.2)

ME= max(px,qy,pZ,qW)~e22~88.

If (4.3)

M= max(pz, qw) > 22.88

then we have, by Lemma 1, j pz f qw ) 2 (max(pz, qw))o.76.

Without loss of generality we may assume ItJ(sqy. Then M=max(pz,qW)s

Ipr*qW)1’0.76<(pWqY+

1)1’o.76s(q2y+ 1)““.76.

If q2y5e16, then Me16, then, by (4.1), Ml(q2y+

1)“o.‘6
Hence z logp< 2.64 log(c,,,z). By Lemma 4, 2.64

e z<----

52

-

e-l

logp

log )2.64% logp

P

_ __: ) y4

P

logp *

~20,

By (3.2), y,.~26,

whence

which contradicts the assumption (4.3). This proves (4.2). Note that, by (2.1), (4.1) and (3.4), IpZ+qWj spxqy+ (4.4)

I I

1 ‘cP,J~,,pwz+

1

8207wz+ 1 if (p,q) or (q,p)=(lOl, i 2323wz+ 1 in all other cases.

181),

By an extensive computer search we find that all unordered pairs@‘, qW satisfying 215< max(pz, qW) 5 e22.88and (4.4) are given by (221, 1273), (216, 1272), (1912,215), (193*,2l’) and (194, 76) and that in all these cases the minus sign in (4.4) holds. Since in none of these cases pz - qWf 1 is composed of only primes p and q, we conclude that max(pz,qW)s2 I5. Thus the assertion of Theorem 1 is true in Case 1. 2. x>zory>w. Without loss of generality we assume z
CASE

qWrpXqY-pz-lr

1-i (

Then, by (2.1),

pxqy-1, >

whence WL y. We see from (2.1) that we can assume w> 0. It follows from (2.1), by Lemma 2, that (4.5)

PZ~CJ$,W,

4 y< -cc&.

We first show that (4.6)

M:=max(px,qy,pZ,qW)seeB+8.22

where p is given in Table A. Note that /3+ 8.22> 18. If max($, qWsY)zeB, then Lemma 1 implies Itiqyt-qwl

=qYlpx-+q w-yl zqY(max($

?qw-y))o.76

2 (max(pYqy, qw))o.76.

Hence (4.7) If pzse6, (4.8)

max(pYqy qW)r JpXqy*qW11’o.76zs(pz+ 1)1’o.76. then Me6, then, by (4.7) and (4.Q max(pYqy, qW)4p1.3221 (c~,~w)1.32.

Hence wlogq~1.3210g(cP,,w). By Lemma 4 and (3.2), we obtain 53

wlogq<

1.32~~ 1.32~ log 2 e-l log 4

< 11.52.

Using (4.8) and (3.3) we find M
M=max(px,qW)
If M<2eP, (4.10)

then (4.6) holds. If MIc,,,ePz,

then

zlogp~p+log(c,,,z),

whence, by Lemma 4, ps22.88 zlogps

- e e-l

and (3.2),

P

log -e C4.P544.48. logp

On substituting this bound in (4.10) and using Table A and (3.2) we find the improved bounds z logps

19.64 if (p,q)=(41,29), (19,137), (53,97), (181,101), 30.79 in all other cases. t

By using (4.9) we obtain that (4.6) also holds in the remaining case. We derive from (4.5) and (4.6) that PZ%4

W’Cpq>

P+8.22 lwq

.

Hence we have, by (3.5)

(4.11)

841 512 343 pz5 243 p if i 169

if (p, q) = (29,41) if (p, q) = (2,127) if (p,q)=(7,19) if (p, q) = (3,163) 173Ip1199 in all other cases.

We may assume that max(p$ qw) > 215. Then we have, by (2. l), that p”qy - 4”’ divides pz f 1. Since qy 1pz f 1, it follows from (4.11) that

(4.12)

q’s

1

127 if (p, q) = (2,127) 128 97 85 if otherwise. p (p,q)=(127,2) = 179 or 193

Since $-- q”-y divides pz f 1, we obtain Theorem 3, that (4.13)

that

IpX- qweyj I

842 and, by

max($, qwmy)5 2002.

By a computer search after all tuples (p, q,x, y, z, w) satisfying 1$qy - q”‘j = 1, max(p”, qw)>215, (4.1 l), (4.12) and (4.13) we complete the proof of 0 Case 2.

pZf

54

PROOF OF THEOREM

We have x>O,

2

y> 0. Hence,

2,

qW5 cq$x.

PZQ,,,Y,

(4.14)

by (2.2) and Lemma

We first show that pXseP with p given in Table A. If pX>eS, then, by Lemma 1rJ: f qy 12 max(d:

1,

qy)o.76 =p”.7&y,

whence pxs Ipx~qy11.32((pz+qw)1.32. By (4.14) qy
p*I(2cP,#32,

then ylogq~1.3210g(2cP,,y) whence,

by Lemma

4 and (3.3),

2.64~ y log q 5 2.09 log d log 4

< 12.98.

This yields, by (4.15) and (3.2), pX%e10.59 unless (p, q) = (137,19). case p” < 1372.3, whence pXs 1372 < e10.59. If

In the latter

pX< (2c,,,x)‘=, then we obtain

p”< e10.59 in the same way. In both cases pXreB.

Note that IpX*qYI = IpZ-+qWI‘cp.Qy+cQ,px. By ylog q
we find

IpXkqyI = jpzkq”‘sb($

+ $),

which is less than 5000 in view of Table A and (3.3). Suppose of (2.2) with max(d: qy) > 2 I5. Then holds and, by Theorem 3, (4.17) Hence, (4.18)

we have a solution

on the left side of (4.16) the minus

sign

max(p”, qy) 5 1992. by (4.14) and (3.2), pz+qw5

5L + c4,p logmax(ti,qY)< ( log 4 logp >

1991.

A search shows that there are no tuples (p, q,x, y, z, W) satisfying JpZ*qWI >o, 215< max(px, qy) < 1992 and (4.18).

$-

qy = 0 55

5. EXAMPLES

In the following examples we search for non-degenerate negative integers x, y, 2, w.

solutions in non-

a) The equation 3’. 11”s 3’= ll”+ 1 has exactly three solutions zw> 0, namely (l,O, 2, l), (2,0,1,1) and (2,2,5,3).

with

Observe that y < w and max(3x, 3’) < 11”‘. According to Theorem 1 we have llw1215 whence ~14. Hence yr3, max(x,z)s8, min(x,@s2. If w is even, then x=0, 3’=l(mod ll), whence z=5, y=2, w$Z. If w=l, then (x, y, z) = (2,0,1) or (LO, 2). If w = 3, then (x, y, z) = (2,2,5). PROOF.

b) The equation 3x+13Y+3z=13w subject solutions, namely (l,O, 2,1) and (2,0,7,3).

to XIZ

has exactly

two

Observe that y< w, max(3x, 3”) < 13!+‘.According to Theorem 2 we have ~54. It follows from y< w, xsz that (w,z)=(4,9) or (3,7) or (2,4) or (1,2). If (w, z) = (4,9), then there are no solutions. If (w,z) = (3,7) then (x, y) = (2,0). If w= 2, then there are no solutions. If w = 1, then (x, y,z) = (l,O, 2). PROOF.

Table A.

Values

of p in Lemma

1.

22.88 if (p, q, k, I) = (2,97,33,5), 20.80 if (p, q. k, I) = (2,181,30,4), 20.62 if (p,q,k,1)=(173,19,4,7), 18.82 if (p, q, k, I) = (23,43,6,5), 17.96 if (p,q,k,1)=(13,89,7,4), 17.68 if (p,q,k,l)=(83,19,4,6), 17.17 if (p,q,k,1)=(31,73,5,4), 16.84 if (p,q,k,1)=(29,67,5,4), 15.89 if (p,q,k,1)=(53,199,4,3), 15.57 if (p,q,k,1)=(7,179,8,3), 15.41 if (p,q,k,1)=(47,3,4,14)

or (47,13,4,6),

15.39 if (p,q,k,1)=(13,3,6,14),

P=

15.06if

(p,q,k,[)=(l51,43,3,4),

14.56if

(p,q,k,1)=(2,127,21,3),

14.19 if (p,q,k,l)=(l13,17,3,5), 13.87 if (p,q,k,1)=(2,101,20,3), 13.74 if (p,q,k,1)=(31,97,4,3), 13.47 if (p, q, k, I) = (29,89,4,3), 13.19if

(p,q,k,1)=(3,2,12,19),

12.88 if (p, q, k, I) = (5,73,8,3), 12.83 if (p,q,k,1)=(13,71,5,3), 11.79if (p,q,k,1)=(2,19,17,4), 11.34 if (p,q,k,1)=(17,5,4,7) 11.29 if (p, q, k, I) = (43,5,3,7), 11.15 if (p,q,k,1)=(41,2,3,16), 10.59 otherwise.

56

or (17,43,4,3),

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