Catalan s Equation Has No New Solution with Either Exponent Less Than 10651

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1 Catalan s Euation Has No New Solution with Either Exonent Less Than 065 Maurice Mignotte and Yves Roy CONTENTS. Introduction and Overview. Bounding One Exonent as a Function of the Other 3. An Alication of Inkeri s First Criterion Acknowledgement References We consider Catalan s euation x y = (where all variables are integers and ; are greater than ), which has the obvious solution 9 8 =. Are there others? Alying old and new theoretical results to a systematic comutation, we were able to imrove on recent work of Mignotte and show that Catalan s euation has only the obvious solutions when minf; g < 065. Two crucial tools used are a new result of Laurent, Mignotte, and Nesterenko on linear forms of logarithms, and a criterion obtained by W. Schwarz in INTRODUCTION AND OVERVIEW In 843, Eugene Catalan considered the following uestion Are there airs of consecutive integers that are both owers, other than ( ; 0), (0; ) and (8; 9)? The general oinion, known as Catalan's conjecture, is that the answer is no. Formally, the relevant diohantine euation is x m y n =, with x; y are integers and m; n integers greater than. Of course, we can assume that the exonents are rime numbers, and, ossibly after interchanging the two terms on the left, that x and y are both nonnegative. Excluding the trivial case of x = and y = 0, the euation we are interested in is x y = ; (.) where ; are rime numbers and x; y are integers greater than. The main results toward the verication of Catalan's conjecture are of relatively recent vintage (see [Ribenboim 994] for a more detailed account). An imortant ste was taken by Tijdeman [976], who roved that the roblem is nite using Baker's results on linear forms in logarithms, he showed that c A K Peters, Ltd /96 $0.50 er age

2 60 Exerimental Mathematics, Vol. 4 (995), No. 4 all unknowns are eectively bounded. The same year, Langevin [977] obtained the exlicit bound 0 0 for maxf; g, and enormous bounds for x and y. Since then rogress on linear forms has led to better bounds. Two years ago, it was ossible to rove maxf; g < 0, and now it seems that maxf; g < 0 8 could be roved. However, we shall not ursue uer bounds in the resent aer, but will focus our attention on imroving the known lower bounds on minf; g. The rst result in this direction [Nagell 90] was minf; g 6= 3. Almost half a century elased until Ko Chao roved that minf; g 6= [Ko 965]. The best result ublished since then [Mignotte 994] had been minf; g 3 Here we reort a signicant advance, roving that minf; g 065 This result was obtained thanks to several theoretical advances and a lot of comutation. To exlain our strategy, it is convenient to generalize (.) slightly to x y = "; with " = and x; y > (.) (still assuming ; rime). This is so that we can interchange the roles of (x; ) and (y; ) as needed. The rst theoretical advance, discussed in Section, is a new lower bound for two linear forms in logarithms [Laurent et al.]. Alied to (.) for a xed rime, it leads to an uer bound < max () We have made great eorts to get a good value for this bound, in order to decrease comutation time for the resent work and to hel the future imrovement of uer bounds on maxf; g. In the rocess (Section.) we resent a technical re- nement of the congruences obtained in 964 by Hyyro. Then, for a xed, we have to consider the range < max (). For each air (; ) we have several theoretical tools to attack (.), which in most cases are sucient to eliminate the ossibility of solutions. Secically, for each rime one can dene two sets F () and H() such that a solution of (.) can only exist for F () [ H() The rst set corresonds to Fermat uotients F () = mod Exeriments show that this set is generally very small, but its comutation takes a very long time. In our case, to comute all these sets for < 065 took more than two weeks on a arallel comuter with 3 rocessors. The reason for the strange value 065 is urely technical the rogram was written in C, in double recision, and 065 is the highest value for which we can comute congruences mod with this rogram. The second set H() is related to certain class numbers, and comes from the rst general algebraic criterion on Catalan's euation, obtained by Inkeri [990]. Inkeri's criterion allows us to ut H() = < max () divides h(k 0 ) ; (.3) where h(k) reresents the class number of a number eld K and K 0 Q [ ] if 3 mod 4, = Q [e i= ] if mod 4. The case mod 4 leads to very serious diculties; the class number of Q [e i= ] is not known for 7. There is a way to overcome this roblem Given, and setting h = h Q [e i= ], there are rocedures that outut either the answer \ does not divide h " or \ may divide h ". But these rocedures are very slow. In Aril 993, Mignotte [995] was able to relace the eld K 0 in the revious criterion by K = the subeld of Q [e i= ] of degree d, where d is the maximal ower of in. For many values of the degree of this new eld K is much smaller than, and h(k ) can be easily

3 Mignotte and Roy Catalan s Euation Has No New Solution with Either Exonent Less Than comuted. But there are still dicult examles, like = 57, where this degree is 56. The newest result we use is from [Schwarz 995], to the eect that in (.3) we can relace h(k ) by h (K ), the relative class number of K over K + (that is, the uotient h(k )=h(k + ); here K + is the maximal real subeld of K ). This reresents an enormous rogress from the comutational oint of view one can comute h (K ) for any [Washington 98]. Without this imrovement we had serious comutational diculties to get minf; g > 570, whereas now the most exensive comutational ste is comuting the Fermat uotients. To summarize the discussion so far, we eliminate most ossibilities for (; ) by using the bound < max () and the following fact Criterion.. Let and be odd rime numbers. Let = d l, where l is odd. Let K = K be the subeld of Q [e i= ] of degree d. Denote by h K the relative class number of K over K+. Then (.) has no solution when both of these relations are satised - h K and 6 mod Now suose that, for a given, we want to analyze a value of that does not satisfy Criterion. (that is, F () [ H()). We have two ways of attack. The more natural, and generally uicker, way is to try Criterion. on the air (; ). We illustrate with the rst values of. For = 5, we have max (5) = 0000, F (5) = f077; 40487g, and H(5) =?; we therefore consider = 077 and = 40487, and examine the ossibility of = 5. Since 5 = F (077); 5 = H(077) = f4g; 5 = F (40487); 5 = H(40487) = f79g; we conclude that (.) has no solution when = 5. Similarly, for = 7, we have max (7) = 0000, F (7) = f5g, and H(7) =?; since we already know that = 5 leads to no solution, we conclude that = 7 also leads to no solution. Sometimes this strategy fails; the smallest examle, already noticed by Inkeri [964], is the air (; )=(83; 487), because 487 F (83) and 83 F (487) In such cases, we try to use the following elementary criterion from [Mignotte 993] Criterion.. Let and be odd rime numbers, and let l be a rime number such that l = h +, with h a ositive integer. Let a and b be integers such that a mod l and b mod l. Then (.) has no solution when all the following relations are satised h 6 mod l, h 6 mod l, and ( + ag j ) h 6 mod l for all j f0; ; ; h g, where g is a rimitive root mod l. For all airs (; ) unresolved by the use of Criterion. (with < 065), the use of Criterion. was sucient to show the absence of solutions, excet for the air (903; 8787). This last case could be solved by congruences mod alied to the formulas obtained during the roof of the rst criterion of Inkeri; the details are too technical to be given here. The conclusion of our comutations is, therefore Theorem.3. Catalan's euation x y = ; where and are rimes and x; y > are integers, has no solutions other than 9 8 = when minf; g < 065. The comuted data can be obtained from the authors. In Section we derive the bound < max () that makes the roblem tractable. In Section 3 we resent a result that is not used in the roof of Theorem.3, but shows that the secial case of Catalan's euation with exonents congruent to 3 mod 4 could be simler than the general case.

4 6 Exerimental Mathematics, Vol. 4 (995), No. 4. BOUNDING ONE EXPONENT AS A FUNCTION OF THE OTHER Arithmetical Relations Suose (x; y; ; ) is a solution to Catalan's euation (.). Cassels [960] roved that there exist integers r and s such that y + " = s and x " = r According to [Hyyro 964], there exist also integers a 0 and u 0 such that a = a 0 " and u = u 0 + satisfy x " = a and x " = (ua) (Hyyro gives additional relations satised by these numbers, but we will not need them.) Since u >, we get so that This imlies x " > ( a) ( ) ; x ( ) (.) r = (x ") > x ( ) = = ; whence log r > log (.) This lower bound seems to be new. In any case, it is uite useful for the estimates in the remainder of this section. A Crude Bound It is easy to rove that s 4 = = r and r 4 = = s, and also that the linear form = log log r s " satises 0 < jj 4 r. Let's assume that > max 400 log ; log (.3) Combined with (.), this imlies log jj log r (.4) We shall aly the following result from [Laurent et al.], where, for an algebraic number, h() = log M()=deg is the logarithmic height of (here M() is the Mahler measure of, the denition of which is recalled on age 67). Theorem.. Let, be two multilicatively indeendent algebraic numbers with j j, j j, and let log and log be any determination of their logs. Put = b log b log ; where b and b are ositive integers. Put D = [Q ( ; ) Q ] [R ( ; ) R ] Let K be an integer, let L, R, R, S, S be ositive integers, and let > be a real number. Suose that R S L and R S > (K )L (.5) Put R = R + R, S = S + S, and g = 4 K Y b = (R )b + (S )b Suose also that KL RS ; (.6) k= =(K K) k! ( ) log i + Dh( i ) a i for i = ; ; that the numbers rb + sb, for 0 r R and 0 s S, are airwise distinct, and that K(L ) log (D + ) log KL D(K ) log(b=) gl(ra + Sa ) > 0 (.7) Then we have the lower bound where LSe 0 LSjj=(b ) = max b j 0 j KL+ ; (.8) ; LReLRjj=(b) b

5 Mignotte and Roy Catalan s Euation Has No New Solution with Either Exonent Less Than Before alying Theorem., we aly a corollary of it [Laurent et al., Corollary ], which is weaker but much simler to use. Corollary.. With the notations of Theorem., suose moreover that and are ositive real numbers. Then n log jj 434 D max 4 where log A i max log b 0 +05; D D ; jlog ij D ; h( i) and b 0 = b log A + b log A o log A log A ; We aly the corollary with b =, b =, = r (s ") ; for i = ;, =, and D =. Notice that and are multilicatively indeendent otherwise would be an integer times log, contradicting the trivial estimate 0 < jj <. Notice also that Moreover, jlog j log + jj log( + ) h( ) max log r + log ; log(s + ) max ( + ) log r; log s + ( + ) log r; since s. Clearly, log = h( ) = log. Thus we can aly Corollary. with log A = ( + ) log r and log A = log (Note that to aly Corollary. we only have to choose log A and log A. Then b 0 is dened in terms of b, b, log A and log A. The corollary gives a lower bound for deending only on these revious uantities and on D.) Hence, by (.3), we have b 0 = log + 00 ( + ) log r log We get log jj 434 maxf; log(=log ) + 05g ( + ) log log r Comaring this ineuality with (.4) leads to 44 (+) log maxf; log(=log )+05g In articular, for 7. A Sharer Bound (.9) In this section we assume. We can aly Theorem. with ( ) a = ( + ) + log r 4 and a = ( + ) log. We shall take 7 5. By (.) and (.3), we have a > log, so that a > , a > 436, and a a > Then, to satisfy condition (.5), we take R = ; S = L; R = (K )La =a + ; S = (K )La =a + We suose that 7 L 5 log. We take K = [ La a ] +, where is some real number to be chosen later, satisfying 0 05; thus K > If there exist two integers r 0 and s 0, with jr 0 j < R and js 0 j < S, such that r 0 b + s 0 b = 0, then divides r 0, so that < R 5 L( + ) < log < 00 log ; which contradicts (.3). Hence, the numbers rb + sb ; for 0 r R and 0 s S, are airwise distinct. We have the following general uer bound for b [Laurent et al., Lemma 6]

6 64 Exerimental Mathematics, Vol. 4 (995), No. 4 b (R )b + (S )b ex K log((k 3 )= e) log K + K 6K(K ) Thanks to our hyotheses on L and K, this leads to log b 5+log L( K +)(b a =a +b a =a ) L a a 5+log (+)( K ) log(38 K) log(k ) K +log b 0 log(38 K) K 5 log (+) +log b 0 + K 5 log((+))+log log + (+) + since now b 0 = ( + ) b a + b a ( + ) + log ( + ) log r log K + log(38 K) K log(38 K) ; K ( + ) Now we consider the uantity g of (.6). From the relations we get R = R + R (K )La =a ; S = S + S L + (K )La =a ; gl(ra + Sa ) = 4 L(Ra + Sa ) KL a S + a R 4 L a + L3= (K )a a KL We have and the identity R (K )La =a ; x + y = x y x + y (x + y)x (.0) a S + a R imlies S (K )La =a + L (K )La =a a L a (K )L L + (K )La =a Hence we obtain the lower bound KL a S + a R (K )L a S + a R L 3= (K )a a a L + Plugging this into (.0) gives gl(ra + Sa ) 3 L3= (K )a a + 3 L a Ignoring the last term, we get a L 3 L + (K )La =a a L 3 L + (K )La =a gl(ra + Sa ) 3 L3= (K )a a + 3 a L Besides, since log r > (=) log, we have a ( + ) log a ( + ) log r < ( + ) log ( + )(=) log < + Using these remarks, we see that condition (.7) is satised if, utting = log, we have 0 < K(L ) + (K ) log log(kl) L3= (K )a 3 a a 3 L (K ) + ( + ) + 5 log(( + )) + log log K log(38 K) K (.) Now the right-hand side of (.) is greater than or eual to +, where = K(L )+K log 999 L3= (K )a 3 a (K ) 5 log ( + ) + log log

7 Mignotte and Roy Catalan s Euation Has No New Solution with Either Exonent Less Than and = 5K 0 4 log(kl ) K ( + ) K a 3 L It is easy to verify that is ositive. Indeed, (+) log < 0 7 ; K K ; log(kl ) < 3 log K K K < ; 3 a L K a L 35 La a = L 35 a < 5 log 6 0 log < 0 4 ; so K(5 46) 0 4 > 0. Then, dividing by La a, we see that condition (.7) is satised when (L )+log log(+) log log + log L 0 (.) 3 In such a case, by (.8) and the ineuality R + S K, we get log jj KL log log(kl) Comaring this ineuality with (.4) gives 000 L ( + )( + ) log log Now we can describe the rocedure used to get an uer bound max () for the exonent in (.), when is xed. We rst aly condition (.9) to get a rst uer bound, say 0, for this exonent. Then, for a suitable choice of and L, we use this uer bound to nd a value for which (.) holds. Then (.) gives an uer bound 0. If < 0 we reeat this rocess using the new uer bound and some choice of and L (ossibly the same as before), which gives an uer bound. We continue in this way, and sto after a certain number of tries, obtaining a value. Finally we take max () = max log ; 400 log ; ; in order to resect (.3). Notice that max () = log for 53. Since = log, condition (.) is euivalent to L log +log + log log + log 3 L 0; where we ut = 5 log 999. This can also be written as L( log ) 3 log + + log log We choose = 3 log ; then the revious condition becomes L 3 log 3 log + + log log For = 9 (so that = 09 [0; 05]), we nd that this ineuality holds if L 06388(log log log ); and we can take L = [06388 log(=log ) + 49] (We verify the condition 7 L 5 log. From (.3), we have L log > 7 Put z = =log ; then (.3) imlies z > 407 and maxf; log(=log ) + 05g < 84 z; thus (.9) gives which leads to and z ( + ) log z; log z < log 87 + log( + ) + log log z log z < 746 log 87 ( + )

8 66 Exerimental Mathematics, Vol. 4 (995), No. 4 Then an elementary numerical study shows that L < 4 log for. This ends the verication.) Thus we get (+) log(=log )+49 log ; which imlies 769 (+) log(=log )+333 log (.3) for ; thus 77 log(=log ) log (.4) for On the range < 065, we have comuted the best ossible value max () obtained by Theorem.. Ineuality (.4) is given as a reference for ossible further comutations. Examle for < 0 4 we have max () < AN APPLICATION OF INKERI S FIRST CRITERION We now rove a result that is not used in the roof of Theorem.3, but shows that the secial case of Catalan's euation with exonents congruent to 3 mod 4 could be simler than the general case. Instead of (.) we will work with the euation x y =, where > > are ositive integers and x, y are (ossibly negative) integers with jxj, jyj >. We will in fact assume that > 50. We recall briey the work in [Inkeri 964]. For rime, with 3 mod 4, suose that a runs over the uadratic residues mod and that b runs over the nonresidues. Put A(X) = Y a where = e i=. Then (X a ); B(X) = Y b (X b ); 4 X X = A(X) B(X) = Y (X) + Z (X); where Y (X) = A(X) + B(X); Z(X) = B(X) A(X) = The olynomials Y and Z have integer coecients. Clearly, deg Y = ( ) and Y (X) = X ( )= + ; L(Y ) L(A) + L(B) (+)= ; where L(P ) denotes the length of the olynomial P (that is, the sum of the modules of its coecients). From the formula on Gauss sums, X a a X b b = ; we see that deg Z = ( 3) and that Z(X) = X ( 3)= + ; L(Z) L(A) + L(B) = (+)= = Now, by Hyyro's theorem (see the beginning of Section ), there exist integers a and u such that jxj = a and jxj = (jxj )u Thus 4u = Y + Z and, if Y = Y=() and Z = Z, then u = (Z + Y )(Z Y ); moreover Y and Z are corime integers [Inkeri 964]. In the uadratic eld Q ( ), this imlies a relation (Z + Y ) = b ; where b is some ideal of this eld. If we assume that does not divide the class number of Q ( ), there exists an algebraic integer, belonging to this eld, such that Z + Y = Hence, = Y and + = Z. Put = jje i, with jj. Then cot() = i + = Z(jxj) Y (jxj)

9 Mignotte and Roy Catalan s Euation Has No New Solution with Either Exonent Less Than Using the revious estimates relative to Y and Z, we get jcot()j < jxj( 3)= + (+)= jxj ( 5)= = jxj ( )= (+)= jxj ( 3)= + (+)= =jxj ; 3jxj jxj (+)= =jxj < since jxj > by an argument like the one leading to (.). Thus there exists an integer k such that the linear form = ki (i) satises jj < jxj We now use [Laurent et al., Theorem 3] Theorem 3.. Let be an algebraic number of modulus that is not a root of unity, let b and b be two ositive integers, and set = b i b log. Put D = [Q () Q ], t max 0; 85 jlog j + Dh() ; n H = max 7; D log b a + b + 46D Then log jj 9tH. In our case we take b = k, b =, = = = e i, and D =. For an algebraic number, let M() denote the Mahler measure of, that is, the roduct ja 0 j dy j= maxf; j j jg; where a 0 is the leading coecient and the j the roots of an irreducible olynomial with integer coecients of which is a root. We have the estimates M() = jj (jz j + jy j ) = or, in terms of the height, x ( )= = = x ( )= ; h() log x o This imlies, with the notation of Theorem 3., that 85 + log x < log x; indeed, (.) says that jxj > =, so that log jxj > 85 because of (.3). Thus we can take t = log x (which imlies t > log ), and then we have H max 7; log (Proof We have 0 < k < and t > log, so D log b a + b D + 35 < log a < 3459; which roves that H maxf7; log + 346g.) Comaring the lower bound of log jj with its uer bound, after some easy simlications, we get 45 H = 45 (maxf7; log + 346g) This uer bound, like (.3) and (.4), is derived here as a reference for ossible further comutations. Note that it is better than (.3) for 3. ACKNOWLEDGEMENT We are very grateful to one of the referees for ointing out several minor mistakes and obscurities and for simlifying considerably the roof of the better bound in Section (ages 63 and following). REFERENCES [Cassels 960] J. W. S. Cassels, \On the euation a x b y =, II", Proc. Cambridge Society 56 (960), 97{03.

10 68 Exerimental Mathematics, Vol. 4 (995), No. 4 [Hyyro 964] S. Hyyro, \ Uber das Catalansche Problem", Ann. Univ. Turku, Ser. AI 79 (964), 0 ages. [Inkeri 964] K. Inkeri, \On Catalan's roblem", Acta Arith. 9 (964), 85{90. [Inkeri 990] K. Inkeri, \On Catalan's conjecture", J. Number Theory 34 (990), 4{5. [Ko 965] Ko Chao, \On the diohantine euation x = y n +, xy 6= 0", Sci. Sinica 4 (965), 457{460. [Langevin 977] M. Langevin, \Quelues alications de nouveaux resultats de van der Poorten", Sem. Delange{Pisot{Poitou (977/78), Paris, Ex. 4, 7 ages. [Laurent et al.] M. Laurent, M. Mignotte, and Y. Nesterenko, \Formes lineaires en deux logarithmes et determinants d'interolation", to aear in J. Number Theory. [Mignotte 99] M. Mignotte, \Sur l'euation de Catalan", C. R. Acad. Sci. Paris, Ser. I 34 (99), 65{68. [Mignotte 993] M. Mignotte, \Un critere elementaire our l'euation de Catalan", C. R. Math. Re. Acad. Sci. Canada 5 (993), 99{00. [Mignotte 994] M. Mignotte, \Sur l'euation de Catalan (II)", Theoret. Com. Sci. 3 (994), 45{ 49. [Mignotte 995] M. Mignotte, \A criterion on Catalan's euation", J. Number Theory 5 (995), 80{83. [Nagell 90] \Des euations indeterminees x +x+ = y n et x +x+ = 3y n ", Nordsk. Mat. Forenings, Skr. () (90), 4 ages. [Ribenboim 994] P. Ribenboim, Catalan's conjecture, Academic Press, Boston, 994. [Tijdeman 976] R. Tijdeman, \On the euation of Catalan", Acta Arith. 9 (976), 97{09. [Schwarz 995] W. Schwarz, \A note on Catalan's euation", Acta Arith. 7 (995), 77{79. [Washington 98] L. C. Washington, Introduction to cyclotomic elds, Sringer, New York, 98. Maurice Mignotte, Universite Louis Pasteur, Centre de Calcul de l'eslanade, 7, rue Rene Descartes, F Strasbourg, France (mignotte@math.u-strasbg.fr) Yves Roy, Universite Louis Pasteur, Centre de Calcul de l'eslanade, 7, rue Rene Descartes, F Strasbourg, France (yr@det-info.u-strasbg.fr) Received December 7, 994; acceted in revised form August 8, 995