Séminaire Bourbaki, Vol. 45, 2002-2003, Exp. 909-923

432 Pages·2003·3.964 MB·French

by  N. Bourbaki

S(cid:19)eminaire BOURBAKI Novembre 2002
55e ann(cid:19)ee, 2002-2003,no 909, p. 1 a(cid:18) 26
CATALAN’S CONJECTURE
[after Mih(cid:21)ailescu]
by Yuri F. BILU
To E.W.
1. INTRODUCTION
In 1844 Crelle’s journal published the following note [13].
Note
extraited’unelettreadress(cid:19)eea(cid:18)l’(cid:19)editeurparMr. E. Catalan,R(cid:19)ep(cid:19)etiteura(cid:18)
l’(cid:19)ecolepolytechnique deParis.
Je vous prie, Monsieur, de vouloir bien (cid:19)enoncer, dans votre recueil, le
th(cid:19)eor(cid:18)eme suivant,quejecrois vrai,bienqueje n’aiepas encorer(cid:19)eussi a(cid:18)
le d(cid:19)emontrercompl(cid:18)etement : d’autres serontpeut-^etreplusheureux:
Deuxnombresentierscons(cid:19)ecutifs, autresque8et9,nepeuvent^etre
des puissances exactes ; autrement dit : l’(cid:19)equation xm(cid:0)yn =1, dans
laquelle les inconnues sont enti(cid:18)eres et positives, n’admet qu’une seule
solution.
Thus, we have the following conjecture.
Conjecture 1.1 (Catalan). | Equation xu yv =1 has no solutions in integers
(cid:0)
x;y;u;v >1 other than 32 23 =1.
(cid:0)
Now,158yearsafter,theconjectureiscompletelyproved. Letusbrie(cid:13)yreviewthe
most important events which lead to the solution of this celebrated problem. This is
not a comprehensivehistoricalaccountof Catalan’sproblem;the lattercanbe found
in Ribenboim’s book [34] and Mignotte’s survey [26].
Seven years after Catalan’s note appeared, Lebesgue [21] proved that equation
xm y2 =1 has no solutions in positive integers x;y;m with m>1. In 1965 Ko
(cid:0)
Chao[18]showedthatequationx2 yn =1hasnosolutionsinpositiveintegersx;y;n
(cid:0)
with n>1 other than 32 23 =1. These two results reduce Catalan’s conjecture to
(cid:0)
the following assertion.
SOCIE(cid:19)TE(cid:19)MATHE(cid:19)MATIQUEDEFRANCE2004
2 Y.F. BILU
Conjecture 1.2. | Equation
(1) xp yq =1
(cid:0)
has no solutions in non-zero integers x;y and odd primes p;q.
Notice that we no longer assume x and y positive. It is convenient, because now
the problem is symmetric: if (x;y;p;q) is a solution, then so is ( y; x;q;p). This
(cid:0) (cid:0)
will be repeatedly used in the sequel.
From now on Conjecture 1.2 will be referred to as Catalan’s conjecture and (1) as
Catalan’s equation.
Cassels [12] discovered important arithmetical properties of solutions of Catalan’s
equation. Hisresults(seeProposition2.1)areindispensableinmostofthesubsequent
works on Catalan’s equation.
In1976Tijdeman [37]madeabreakthrough. UsingBaker’stheory,heprovedthat
the exponents p and q are bounded by an explicit absolute constant. Together with
theclassicalresultofBaker[6]thisimpliesthat x and y areboundedbyanexplicit
j j j j
absolute constant as well, and Catalan’s problem is thereby decidable.
In a di(cid:11)erent direction, Inkeri [16,17] and others obtained algebraic criteria of
solubility of (1) in terms of the exponents p and q. In the nineties, Mignotte and
Roy used Inkeri-type criteria, Tijdeman’s argument and electronic computations to
obtain tight lower and upper bounds for p and q. (Upper bounds were also obtained
by Blass et al. [10] and O’Neil [32].) By 2000, it was provedthat p and q lie between
107 and 1018. See [29] for more precise results and a survey of this period.
In 1999 Preda Miha(cid:21)ilescu enters the scene. In his (cid:12)rst paper [29] he drastically
re(cid:12)ned Inkeri’s criterion. And quite recently, after several unsuccessful attempts, he
(cid:12)nally settled [30] Catalan’s conjecture:
Theorem 1.3 (Miha(cid:21)ilescu). | Conjecture 1.2 is true.
The present paper contains a reasonablyself-contained proof of this result.
Plan of the paper. | InSection 2we recallCassels’relationsand derive theirimme-
diateconsequence,inparticular,Hyyr(cid:127)o’slowerboundsfor x and y . InSection3we
j j j j
very brie(cid:13)y review algebraic criteria for Catalan’s equation in terms of p and q, and
proveMiha(cid:21)ilescu’s\doubleWieferich"criterion. InSection4weusebinarylogarithmic
forms, Tijdeman’s argument, and computations by Mignotte and Roy to show that
p 1modq. Section5containsgenerallemmas. InSection6Theorem1.3isreduced
6(cid:17)
to three more technical statements, which are proved in the three (cid:12)nal section.
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(909) CATALAN’S CONJECTURE 3
Acknowledgements. | My deepest gratitude goes to Hendrik W. Lenstra and Yann
Bugeaud,whocarefullyreadthemanuscriptandsuggestednumerouscorrectionsand
improvements. I am indebted to Yann Bugeaud, Andrew Glass, Guillaume Hanrot,
Maurice Mignotte and Preda Miha(cid:21)ilescu for explaining to me various results from
Sections 2 and 4 and other useful discussions. I also thank Bruno Angl(cid:18)es, John
Coates, Gabi Hecke, Shanta Laishram, Hendrik W. Lenstra, Tauno Mets(cid:127)ankyl(cid:127)a and
GisbertWu(cid:127)stholz,whodetectedinaccuraciesinpreviousversionsofthisnote. Finally,
I thank Denis Benois and Leonid Positselski for a tutorial in commutative algebra.
1.1. Notation
In the sequel we assume, unless the contrary is indicated explicitly, that x;y are
non-zero integers and p;q are odd prime numbers satisfying
(2) xp yq =1:
(cid:0)
Aswehadalreadynoticed,(2)impliesthat( y)q ( x)p =1,andallthestatements
(cid:0) (cid:0) (cid:0)
below remain true with x;y;p;q replaced by y; x;q;p.
(cid:0) (cid:0)
We denote by (cid:16) a primitive p-th root of unity and put
K =Q((cid:16)); G=Gal(K=Q):
The principal ideal (1 (cid:16)) will be denoted by p. Recall that it is a prime ideal of K
(cid:0)
and that (p)=pp(cid:0)1.
More speci(cid:12)c notation will be introduced at the appropriate places.
2. CASSELS’ RELATIONS AND LOWER ESTIMATES FOR x
j j
AND y
j j
Cassels [12] proved that q x and py. More precisely, he established the following
j j
relations.
Proposition 2.1 (Cassels). | There exist a non-zero integer a and a positive inte-
ger v such that
(3) x 1=pq(cid:0)1aq; y=pav;
(cid:0)
xp 1
(4) (cid:0) =pvq;
x 1
(cid:0)
and, symmetrically, there exist a non-zero integer b and a positive integer u such that
(5) y+1=qp(cid:0)1bp; x=qub;
yq +1
(6) =qup:
y+1
SOCIE(cid:19)TE(cid:19)MATHE(cid:19)MATIQUEDEFRANCE2004
4 Y.F. BILU
The following consequence is crucial.
Corollary 2.2. | The number (cid:21):=(x (cid:16))=(1 (cid:16)) is an algebraic integer. The
(cid:0) (cid:0)
principal ideal ((cid:21)) is a q-th power of an ideal of the (cid:12)eld K.
Proof. | Sincep(x 1)by(3),theprimeidealp=(1 (cid:16))dividesx (cid:16),butp2 does
j (cid:0) (cid:0) (cid:0)
not. Hence (cid:21) is an algebraic integer, not divisible by p, and the same is true for its
conjugates(cid:21)(cid:27),where(cid:27) G. Identity(1 (cid:16)(cid:27))(cid:21)(cid:27) (1 (cid:16)(cid:28))(cid:21)(cid:28) =(cid:16)(cid:28) (cid:16)(cid:27) impliesthat
2 (cid:0) (cid:0) (cid:0) (cid:0)
fordistinct(cid:27);(cid:28) G,thegreatestcommondivisorof(cid:21)(cid:27) and(cid:21)(cid:28) divides((cid:16)(cid:28) (cid:16)(cid:27))=p.
2 (cid:0)
Hence the numbers (cid:21)(cid:27) are pairwise co-prime.
Now rewrite (4) as (cid:21)(cid:27) =vq. Since the factors are pairwise co-prime, each
(cid:27)2G
principal ideal ((cid:21)(cid:27)) is aQq-th power of an ideal.
Cassels’ relations imply various lower estimates for the variables x and y in terms
of p and q. For instance, (3) and (5) immediately yield
(7) x >pq(cid:0)1 1;
j j (cid:0)
(8) y >qp(cid:0)1 1;
j j (cid:0)
and this can be re(cid:12)ned without much e(cid:11)ort.
Hyyr(cid:127)o [15] obtained an estimate of a di(cid:11)erent kind: x >q(2p+1)(2qp(cid:0)1+1)
j j
(and similarly for y ). Since Hyyr(cid:127)o’s paper is not easily available, I prove below a
j j
slightly weaker estimate, which is totally su(cid:14)cient for our purposes. It is an easy
consequence of the following proposition.
Proposition 2.3. | If p does not divide q 1 then qp(cid:0)2 (u 1).
(cid:0) (cid:0)
(cid:12)
(cid:12)
Proof. | Rewriting (6) as
( y)q(cid:0)1 1 + ( y)q(cid:0)2 1 + +( y 1)=q(up 1);
(cid:0) (cid:0) (cid:0) (cid:0) (cid:1)(cid:1)(cid:1) (cid:0) (cid:0) (cid:0)
(cid:0) (cid:1) (cid:0) (cid:1)
we deduce that (y+1) (q(up 1)). Now (5) implies that up 1modqp(cid:0)2. Since p
j (cid:0) (cid:17)
does not divide the order qp(cid:0)3(q 1) of the multiplicative group modqp(cid:0)2, this im-
(cid:0)
plies that u 1modqp(cid:0)2.
(cid:17)
Corollary 2.4. | We have x >qp(cid:0)1.
j j
Proof. | If p(q 1) then p<q and the result follows from (7). If p does not di-
j (cid:0)
videq 1then qp(cid:0)2 (u 1),and,sinceuispositive,thisimpliesu>qp(cid:0)2+1. Since
(cid:0) (cid:0)
x=qub, we have x(cid:12)>qu>qp(cid:0)1+q, better than wanted.
j j(cid:12)
Remark 2.5. | This version of Hyyr(cid:127)o’s argument is due to Mignotte and Bugeaud.
It was kindly communicated to me by Yann Bugeaud. Using more advanced tools,
Miha(cid:21)ilescu [30, Appendix A] obtained a much sharper estimate x > q2p(cid:0)2=2 4.
j j
(cid:0) (cid:1)
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(909) CATALAN’S CONJECTURE 5
3. ALGEBRAIC CRITERIA
Using Cassels’ relations and some algebraic number theory, one may get various
algebraic criteria of solvability of Catalan’s equation with given exponents p and q.
The most famous criterion is due to Inkeri [16,17]:
Theorem 3.1 (Inkeri). | With the notation of Subsection 1.1, put K =Q(p p)
p
(cid:0)
if p 3mod4 and K =K if p 1mod4. Then either pq(cid:0)1 1modq2 or q divides
p
(cid:17) (cid:17) (cid:17)
the class number of the (cid:12)eld K .
p
It will be explained in Subsection 4.4 how algebraic criteria of this kind, together
with electronic computations, allow one to obtain lowerbounds for p and q.
Re(cid:12)nements of and supplements for Inkeri’s criterion were suggested by Mignotte
[25], Schwarz [35] and others; see [26] for a survey of these results. I would es-
pecially mention the paper by Bugeaud and Hanrot [11], which strongly in(cid:13)uenced
Miha(cid:21)ilescu’s work.
Veri(cid:12)cation of Inkeri’s criterion for a given pair (p;q) requires computing certain
class numbers, which seriously a(cid:11)ects its computational e(cid:14)ciency. Miha(cid:21)ilescu [29]
madeamajorstepforward,showingthatthe classnumberconditioncanbe omitted.
Theorem 3.2 (Miha(cid:21)ilescu). | For any solution of (x;y;p;q) of (2) we have q2 x
j
and
(9) pq(cid:0)1 1modq2:
(cid:17)
Congruence(9)(calledWieferich’s relation)willbeusedinSection4toprovethat
p 1modq. Relation q2 x is crucial in the proof of Theorem 6.3.2.
6(cid:17) j
By symmetry, one has qp(cid:0)1 1modp2. Pairs (p;q), satisfying this and (9) are
(cid:17)
called double Wieferich pairs. Only six such pairs are currently known:
(2;1093);(3;1006003);(5;1645333507);(83;4871);(911;318917);(2903;18787):
I sketch the proof of Theorem 3.2, because it is very instructive and can serve as a
goodmodelofthe muchmoreinvolvedproofofTheorem1.3. See[24,33]fordi(cid:11)erent
proofs.
3.1. Proof of Theorem 3.2
For a 1;2;:::;p 1 let (cid:27) be the element of G=Gal(K=Q) be de(cid:12)ned by
a
2f (cid:0) g
(cid:16) (cid:16)a. In the group ring Z[G] consider elements
7!
p(cid:0)1
(cid:2) = ac=p (cid:27)(cid:0)1 (c=1;2;:::;p 1):
c b c a (cid:0)
Xa=1
In particular, (cid:2) =0 and (cid:2) =(cid:27) + +(cid:27) . Ideal =((cid:2) ;(cid:2) ;:::;(cid:2) )
1 2 (p+1)=2 p(cid:0)1 1 2 p(cid:0)1
(cid:1)(cid:1)(cid:1) I
ofZ[G]iscalledtheStickelbergerideal. ItsmainpropertyistheStickelbergertheorem:
SOCIE(cid:19)TE(cid:19)MATHE(cid:19)MATIQUEDEFRANCE2004
6 Y.F. BILU
any (cid:2) annihilates the class group of K. That is, for any ideal a of K and any
2I
(cid:2) , the ideal a(cid:2) is principal. See [39, Section 6.2] for the details.
2I
Let(cid:19)=(cid:27) bethecomplexconjugation. Miha(cid:21)ilescuprovesthefollowingassertion.
p(cid:0)1
Proposition 3.1.1. | For any (cid:2) (1 (cid:19)) , the element (x (cid:16))(cid:2) is a q-th power
2 (cid:0) I (cid:0)
in K.
Proof. | Write (cid:2)=(1 (cid:19))(cid:2)0, where (cid:2)0 . Put (cid:21):=(x (cid:16))=(1 (cid:16)). By Corol-
(cid:0) 2I (cid:0) (cid:0)
lary 2.2 the principal ideal ((cid:21)) is a q-th power: ((cid:21))=aq. By the Stickelberger the-
orem a(cid:2)0 is a principal ideal, say, ((cid:11)). It follows that (cid:21)(cid:2)0 =((cid:11))q, or (cid:21)(cid:2)0 =(cid:17)(cid:11)q,
(cid:16) (cid:17)
where (cid:17) is a unit of K. We obtain
1 (cid:16) (cid:2)0 (cid:17) (cid:11) q
(10) (x (cid:16))(cid:2) = (cid:0) :
(cid:0) (cid:18)1 (cid:16)(cid:19) (cid:17) (cid:16)(cid:11)(cid:17)
(cid:0)
Since (cid:17) is a unit, (cid:17)=(cid:17) is a root of unity(1) . The quotient (1 (cid:16))=(1 (cid:16)) is a root of
(cid:0) (cid:0)
unity as well. Thus, (x (cid:16))(cid:2) is a q-th powertimes a root of unity. Since any root of
(cid:0)
unity in K is a q-th power, so is (x (cid:16))(cid:2).
(cid:0)
Proof of q2 x. | Since(1 (cid:16)x)(cid:2) isequalto(x (cid:16))(cid:2) timesarootofunity,itisaq-th
j (cid:0) (cid:0)
power as well. On the other hand, q x implies that (1 (cid:16)x)(cid:2) 1modq. Since q is
j (cid:0) (cid:17)
unrami(cid:12)edinK,thisimpliesthat(1 (cid:16)x)(cid:2) 1modq2 (cf. Proposition5.3.1below).
(cid:0) (cid:17)
However, if (cid:2)= n (cid:27), then a quick calculation shows that
(cid:27)2G (cid:27)
P
(1 (cid:16)x)(cid:2) 1 x n (cid:16)(cid:27)modq2:
(cid:27)
(cid:0) (cid:17) (cid:0)
(cid:27)X2G
It follows that either q2 x or q n (cid:16)(cid:27). In the latter case q n for any (cid:27) G.
j j (cid:27)2G (cid:27) j (cid:27) 2
However, this is not true if, for iPnstance,
(cid:2)=(1 (cid:19))(cid:2) = (cid:27)(cid:0)1 (cid:27)(cid:0)1 +(cid:27)(cid:0)1 + +(cid:27)(cid:0)1 :
(cid:0) 2 (cid:0) 1 (cid:0)(cid:1)(cid:1)(cid:1)(cid:0) (p(cid:0)1)=2 (p+1)=2 (cid:1)(cid:1)(cid:1) p(cid:0)1
Thus, q2 x. .
j
Proof of (9). | This is just an elementary exercise. Since q2 x, the (cid:12)rst equality
j
in (3) implies that
(11) pq(cid:0)1aq 1modq2:
(cid:17)(cid:0)
Since pq(cid:0)1 1modq, we have aq 1modq, which implies aq 1modq2, which,
(cid:17) (cid:17)(cid:0) (cid:17)(cid:0)
together with (11), implies (9).
(1)Itisanalgebraicinteger,andforany(cid:27)2Gwehavej((cid:17)=(cid:17))(cid:27)j=1
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(909) CATALAN’S CONJECTURE 7
4. LOGARITHMIC FORMS, TIJDEMAN’S ARGUMENT AND THE
RELATION p 1modq
6(cid:17)
As I mentioned in the introduction, Tijdeman [37] applied Baker’s theory of
logarithmicformtoestablishane(cid:11)ectiveupper boundforthe solutions,reducingthe
problem to a (cid:12)nite computation. In this section we use Tijdeman’s argument and
electronic computations due to Mignotte and Roy to prove the following important
theorem.
Theorem 4.1. | Let (x;y;p;q) be a solution of (2). Then p 1modq.
6(cid:17)
The relation p 1modq is indispensable for Miha(cid:21)ilescu’s proof. It is repeatedly
6(cid:17)
used in Section 6 and in the proof of Theorem 6.3.2. A reader ready to take Theo-
rem 4.1 for granted may skip the rest of this section.
Whenwritingthissection,Ipro(cid:12)tedalotfromhelpfulexplanationsandsuggestions
of Maurice Mignotte and Andrew Glass.
4.1. Logarithmic forms
In this subsection we recall Baker’s lower bound for logarithmic forms
(cid:3)=b log(cid:11) + +b log(cid:11) :
1 1 n n
(cid:1)(cid:1)(cid:1)
Here b ;:::;b are non-zero integers and (cid:11) ;:::;(cid:11) are usually algebraic numbers.
1 n 1 n
To avoid unnecessary technicalities, we shall assume that (cid:11) ;:::;(cid:11) are positive ra-
1 n
tional numbers, distinct from 1. Thisistotallysu(cid:14)cientforapplicationsinCatalan’s
problem.
De(cid:12)netheheightofarationalnumber(cid:11)=(cid:22)=(cid:23) (where(cid:22)and(cid:23) arerelativelyprime
integers) by h((cid:11))=logmax (cid:22); (cid:23) . Assume that (cid:3)=0. Then it is rather easy to
fj j j jg 6
bound (cid:3) frombelow. Indeed,e(cid:3) 1isanon-zerorationalnumberwithdenominator
j j (cid:0)
bounded by e(h((cid:11)1)+(cid:1)(cid:1)(cid:1)+h((cid:11)n))B, where
B =max b ;:::; b :
1 n
fj j j jg
It follows that
(12) (cid:3) e(cid:0)(h((cid:11)1)+(cid:1)(cid:1)(cid:1)+h((cid:11)n))B;
j j(cid:29)
where here and below in this subsection the positive constants implied by O(),
(cid:1)
and are absolute and e(cid:11)ective.
(cid:28) (cid:29)
However,(12)istooweakforapplications: oneneedso(B)intheexponent. Suchan
estimatewasobtainedbyGelfond[14]forn=2andbyBaker[2{5]inthegeneralcase.
Baker’sinequality belongs to the top arithmetical results of the twentieth century.
The modern estimate [7,22,23,38]is of the form
(13) (cid:3) >e(cid:0)c(n)h((cid:11)1)(cid:1)(cid:1)(cid:1)h((cid:11)n)logB
j j
SOCIE(cid:19)TE(cid:19)MATHE(cid:19)MATIQUEDEFRANCE2004
8 Y.F. BILU
(provided (cid:3)=0.) See the recent volume [40] for the history of the subject and the
6
present state of art.
When one wants to be explicit, the numerical value of the constant c(n) becomes
vital. For growingn, the best result is due to Matveev [22,23],who showed that one
may take c(n)=cn with an explicit absolute constant c.
However,inCatalan’sproblemoneuses(13)onlywithn=2andn=3. Therefore
itispracticaltohavespecialboundsforthesetwocases,whicharenumericallysharper
than the general bound (13). Such bounds were obtained by Laurent, Mignotte and
Nesterenko[20]forbinaryformsandbyBennettet al. [8]forternaryforms. Hereisa
simpli(cid:12)ed form of the Laurent-Mignotte-Nesterenkoresult (see Corollary 2 from [20,
Section 2]), to be used in Subsection 4.3 below.
Proposition 4.1.1. | Let (cid:11) ;(cid:11) be multiplicatively independent positive ratio-
1 2
nal numbers and b ;b positive integers. Let A ;A be real numbers satisfying
1 2 1 2
A >max h((cid:11) );1 fori=1;2. PutB =b =A +b =A and(cid:3)=b log(cid:11) b log(cid:11) .
i i 1 2 2 1 1 1 2 2
f g (cid:0)
Then
(14) log (cid:3) > 24:34(max logB+0:14; 21 )2A A :
1 2
j j (cid:0) f g
This is asymptotically weaker than (13) when B grows (because logB is replaced
by (logB)2), but for small B inequality (14) is very sharp numerically.
Idonotformulatethe resultof[8], becauseitisveryinvolvedandwillnotbeused
here.
4.2. An informal introduction to Tijdeman’s argument
In this subsection we assume that p>q. In Catalan’s problem, the most obvious
logarithmic form to try is (cid:3)=plog x qlog y . The upper estimate is obvious:
j j(cid:0) j j
(cid:3) 6 x(cid:0)p. The lower estimate coming from (13) is (cid:3) >e(cid:0)O(plogjxjlogjyj), and
j j j j j j
comparing the two estimates does not yield any interesting consequence.
Tijdeman’s [37] brilliant idea was to use (cid:3)=qlog y+1 plog x. Upper bound
j j(cid:0) j j
isnowslightlyworse: (cid:3) q y (cid:0)1. Forthe lowerbound, weuseCassels’relations(6)
j(cid:28) j j
toobtain(cid:3)=plog(cid:11) qlogq,where(cid:11)=(q b)q(cid:0)1u(cid:0)1(recallthatu>0). Itiseasyto
(cid:0) j j
show(seeSubsection4.3)thath((cid:11))=log u +O(1)6(q=p)log y +O(1). Now(13)
j j j j
implies that (cid:3) >e(cid:0)O((q=p)logjyjlogqlogp), which, compared with the lower estimate,
j j
implies that
(15) p qlogqlogp:
(cid:28)
If (14) is used instead of (13), then one obtains the slightly weaker inequality
(16) p qlogq(logp)2:
(cid:28)
Similarly, using
(cid:3)=qlog y+1 plog x 1 =pqlog(cid:12) qlogq+plogp
j j(cid:0) j (cid:0) j (cid:0)
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(909) CATALAN’S CONJECTURE 9
with (cid:12) =bq=ap, one obtains the estimate
(17) q (logp)2logq:
(cid:28)
Together with (15) this implies an e(cid:11)ective upper bound for p, as wanted.
As I already mentioned in Subsection 4.1, Tijdeman’s argument does not require
thefullstrengthofBaker’sinequality. Oneneedsalowerboundforbinarylogarithmic
forms to obtain (15) and a lower bound for ternary logarithmic form to obtain (17).
Langevin [19] made Tijdeman’s work explicit by proving that p;q 610110. This
bound has been re(cid:12)ned several times until O’Neil [32] (see also [10]) proved that
p63:2 1017 and q 62:6 1012, and Mignotte [29]announcedthat p67:8 1016 and
(cid:1) (cid:1) (cid:1)
q 67:2 1011. Mignotte used the already mentioned bounds for binary and ternary
(cid:1)
logarithmic forms from [20] and [8], respectively.
4.3. Explicit Tijdeman’s inequality
InthissubsectionweapplyProposition4.1.1toobtainanexplicitanalogueof(16).
Proposition 4.3.1. | For any solution of (2) we have
2
p+1
(18) p624:34q max log +0:14; 21 logq:
(cid:18) (cid:26) logq (cid:27)(cid:19)
Inequality(18)willbeusedinSubsection4.5. Itislesssharpthanthecorrespond-
ing results from [27] and [10], but easier to prove and su(cid:14)cient for our purposes.
Proof of Proposition 4.3.1. | We may assume that
(19) p>10000qlogq;
and, in particular, p>q, since otherwise (18) holds trivially.
As indicated in Subsection 4.2, we will compareupper and lowerestimates for the
quantity
(cid:3)=qlog y+1 plog x =plog(cid:11) qlogq;
j j(cid:0) j j (cid:0)
with (cid:11)=(q b)q(cid:0)1u(cid:0)1, where b Z and u Z are de(cid:12)ned in Proposition 2.1.
>0
j j 2 2
The upper estimate is trivial. Rewriting Catalan’s equation (2) as
plog x =qlog y +log(1+y(cid:0)q);
j j j j
we obtain
(20) (cid:3)=qlog 1+y(cid:0)1 log 1+y(cid:0)q :
(cid:0)
(cid:0) (cid:1) (cid:0) (cid:1)
Since log(1+t) 62t for t 61=2, this implies that
j j j j j j
(21) (cid:3) 62y (cid:0)q+2q y (cid:0)1 63q y (cid:0)1;
j j j j j j j j
and (cid:3) <1 by (8). Equality (20) implies also that (cid:3)=0: the (cid:12)rst term always
j j 6
dominates overthe second one.
SOCIE(cid:19)TE(cid:19)MATHE(cid:19)MATIQUEDEFRANCE2004
10 Y.F. BILU
For the lower bound, let us estimate h((cid:11)). Since
log (q b)q(cid:0)1 =logu+((cid:3)+qlogq)=p6logu+1;
j j
(cid:0) (cid:1)
we have h((cid:11))6logu+1. Also, q and (cid:11) are multiplicatively independent: other-
wise, (cid:3) would have been a multiple of logq, contradicting the previously established
inequality 0< (cid:3) <1.
j j
Thus, we are in a position to use Proposition4.1.1. We obtain
(22) log (cid:3) > 24:34(max logB+0:14;21 )2(logu+1)logq
j j (cid:0) f g
with B =p=logq+q=(logu+1). Proposition2.3 and (19) imply that
(23) u>qp(cid:0)2 >e9000q(logq)2:
Hence B 6(p+1)=logq . Substituting this into (22) and combining the resulting
inequality with (21), we obtain
2
log y p+1 1 log(3q)
(24) j j 624:34 max log +0:14;21 logq 1+ + :
logu (cid:18) (cid:26) logq (cid:27)(cid:19) (cid:18) logu(cid:19) logu
Further, (6) implies that q y q(cid:0)1 >qup, whence
j j
log y
(25) p6(q 1) j j
(cid:0) logu
2
p+1 1 (q 1)log(3q)
624:34(q 1) max log +0:14; 21 logq 1+ + (cid:0) :
(cid:0) (cid:18) (cid:26) logq (cid:27)(cid:19) (cid:18) logu(cid:19) logu
Using(19)and(23),oneeasilyshowsthattheright-handsideof(25)doesnotexceed
the right-hand side of (18). The proposition is proved.
4.4. Lower bounds for p and q
One can bound exponents p and q from below, using algebraic criteria (see Sec-
tion3)andelectroniccomputations. ThishasbeenrealizedbyMignotteandRoy[27{
29]. To show that q >Q , one has to verify an algebraiccriterion (Inkeri’sor other),
0
forallpairs(p;q)satisfyingq 6Q ,p>q and(18). Actually,MignotteandRoyused
0
sharper, than (18), inequalities.
With Inkeri-type criteria, Mignotte and Roy managed to provethat
(26) min p;q >105;
f g
using several months of computations. With Miha(cid:21)ilescu’s criterion (Theorem 3.2)
this requiredonlya fewhoursof computations,andwith one monthof computations
they managed to prove that min p;q >107. I am aware about the computations of
f g
Granthamand Wheeler showing that min p;q >3:2 108 but I havenever seen this
f g (cid:1)
result announced in print.
Inequality (26) will be used in Subsection 4.5.
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