Complete solution of parametrized Thue equations

Transcription

1 Complete solution of prmetrized Thue equtions C. Heuberger A. Pethő R. F. Tichy..998 Abstrct We give survey on recent results concerning prmetrized Thue equtions. Moreover, we solve completely the fmily XX Y X Y X + Y Y 4 = ±. Keywords: Diophntine equtions, Symbolic Computtion, Liner Forms in Logrithms Clssifiction: D57, Y50 Introduction Let F Z[X, Y ] be homogeneous, irreducible polynomil of degree n nd m be n integer. Then the diophntine eqution F x, y = m is clled Thue eqution in honour of A. Thue, who proved in 909 []: Theorem. Thue. hs only finitely mny solutions x, y Z. Thue s proof is bsed on his pproximtion theorem: Let α be n lgebric number of degree n nd ε > 0. Then there exists constnt cα, ε, such tht for ll p Z nd q N α p cα, ε q q. n/++ε Since this pproximtion theorem is not effective, Thue s theorem is not. Studying liner forms in logrithms of lgebric numbers, A. Bker could give n effective upper bound for the solutions of such n eqution in 968 []: Theorem. Bker. Let κ > n + nd x, y Z be solution of. Then mx{ x, y } < Ce logκ m, where C = Cn, κ, F is n effectively computble number. Since tht time, these bounds hve been improved; Bugeud nd Győry [6] recently gve the following bound: All three uthors were supported by the Hungrin-Austrin governmentl scientific nd technologicl coopertion. The first uthor ws supported by FWF Grnt 0-MAT. The second uthor ws supported prtly by the Hungrin Ntionl Foundtion for Scientific Reserch Grnt No 679/95.

2 Theorem. Bugeud-Győry. Let B mx{ m, e}, α be root of F X,, K := Qα, R := reg K the regultor of K nd r the unit rnk of K. Let H be n upper bound for the bsolute vlues of the coefficients of F. Then ll solutions x, y Z of stisfy mx{ x, y } < exp C R mx{log R, } R + loghb nd mx{ x, y } < exp C H n log n H log B, with C = r+7 r + 7r+9 n n+6r+4 nd C = n+9 n 8n+. Bombieri nd Schmidt [5] proved tht the number of solutions of with m = ± is On: Theorem.4 Bombieri-Schmidt. There is n bsolute constnt C 0 such tht for ll n C 0 the diophntine eqution F X, Y = ± hs t most 4 n solutions. Up to mybe the constnt 4, this is best possible, since the eqution X n + X Y X Y... nx Y = ± hs t lest the n + solutions ±{,,...,, n, 0, ±}. However, the bounds obtined by Bker s method re rther lrge, thus the solutions prcticlly cnnot be found by simple enumertion. Bker nd Dvenport [] proposed for similr problem method to reduce drsticlly the bound by using continued frction reduction. Pethő nd Schulenberg [6] replced the continued frction reduction by the LLL-lgorithm nd gve generl method to solve in the totlly rel cse for m = nd rbitrry n. Tznkis nd de Weger [] describe the generl cse. Finlly, Bilu nd Hnrot [4] observed tht Thue equtions imply not only one, but r independent liner forms in logrithms of lgebric numbers in the sme very smll size. They were ble to replce the LLL-lgorithm by the much fster continued frction method nd solve Thue equtions up to degree 000. In 990, Thoms investigted for the first time prmetrized fmily of Thue equtions. Since then, the following fmilies hve been studied:. X X Y + XY Y =. Thoms [9] nd Mignotte [] proved tht for 4, the only solutions re 0,,, 0 nd, +, while in the cses 0 4 there exist some nontrivil solutions, too, which re given explicitely in [9].. X X Y + XY Y +. All solutions of this Thue inequlity hve been found by Mignotte, Pethő, nd Lemmermeyer [].. X + X Y + XY Y =. Lee [5] nd independently Mignotte nd Tznkis [4] proved tht for. 0, only trivil solutions exist. Very recently, Mignotte [0] could solve this eqution completely. 4. XX n Y X n b Y ± Y =. This fmily ws investigted by Thoms [0]. He proved tht for 0 < < b nd n b 4.85/b nontrivil solutions cnnot exist. He lso investigted this fmily with n nd n b replced by polynomils in n of degrees nd b, respectively.

3 5. X 4 X Y X Y + XY + Y 4 = ±. This qurtic fmily ws solved by Pethő [5] for lrge vlues of ; Mignotte, Pethő, nd Roth [] solved it completely. 6. X 4 X Y X Y + XY + Y 4 = ± hs been solved for by Pethő [5]. 7. X 4 X Y 6X Y + XY + Y 4 {±, ±4}. This eqution hs been solved by Lettl nd Pethő [6]; Chen nd Voutier [7] solved it independently by using hypergeometric method insted of Bker s method. 8. XX Y X Y X by Y 4 = ±. All solutions of this two-prmetric fmily re known for 0 08 < + < b + log 4, cf. Pethő nd Tichy [7]. The cse b = + will be considered in this pper. 9. Wkbyshi [4] proved tht if x 4 x y + y 4 nd 8 then y. 0. Hlter-Koch, Lettl, Pethő, nd Tichy [] investigted for distinct integers = 0,,..., n nd n integrl prmeter n = the eqution n X i Y ± Y n = ±. i=. XX Y X Y Y 5 = ±. For >.6 0 9, ll solutions hve been found by Heuberger [].. X 6 X 5 Y 5 + 5X 4 Y 0X Y + 5X Y XY 5 + Y 6 {±, ±7} ws investigted by Lettl, Pethő, nd Voutier, they found ll solutions for 89 by hypergeometric methods [7]. For < 89 they used Bker s method [8]. Generl pproch nd Liner Forms in Logrithms of lgebric numbers. Thue equtions In this section, we give short survey on the generl pproch to solve single Thue eqution cf. Gál [0]. In order to keep nottion simple, we only consider the eqution F X, Y = ±, where fx := F X, = n i X i is monic polynomil with rel zeros α,..., α n. Let x, y Z be solution of. We define j {,..., n} by x y αj = min x i {,...,n} y αi. i=0

4 Then we hve y α i α j x α i y + x α j y x α i y nd x α j y = i j x α i y y n i j n α i α j c n, 4 y where c,... denote positive effectively computble constnts depending on K := Qα. For y > c /n, x/y is convergent of α j by Lgrnge s theorem. This yields We hve F X, Y = Y n f yα j α i c y n < x αi y < yα j α i + c n. 5 y X n X = Y n Y Y αi = i= n X α i Y = N K/Q X α Y. Set β i := x α i y for i =,..., n, then β is unit in O := Z[α ]. Thus by Dirichlet s theorem, we obtin i= β = ±ε u... εur r, u,..., u r Z, 6 where ε,..., ε r re fundmentl units of O. Considering 6 for ll conjugtes nd tking logrithms, we get the following system of liner equtions in the u i : log β i = u log ε i + + u r log ε i i j. 7 Using 5, we derive the estimte U := mx u i c mx log β i c log y. 8 i r i j For k l {,..., n} \ {j}, together with 4 nd 5 Siegel s identity r αj α k α j α l x αl y x α k y = αl α k α l α j x αj y x α k y 9 yields αj α k β l α j α l β k = α l α k α l α j β j β k c 4 y n. 0 Using lower bound for liner forms in logrithms of lgebric numbers see section., 6, 0, 4, 5 nd finlly 8 we hve exp c 5 log U < log α j α k α j α l + u log ε l ε k + + u r log ε l r ε k r αj α k βl α j α l β k c 4 y n = expc 6 n log y < expc 6 c 7 U. This estimte cn only hold for U c 8 yielding n upper bound for y by 7 nd 5. 4

5 . Prmetrized fmilies of Thue equtions Given prmetrized fmily of Thue equtions, one hs to perform the sme steps s in the cse of one single Thue eqution using symptotic bounds for the quntities involved. The min extr tool re estimtes of the form U > C g log for some positive constnt C nd some g N. Thoms [0] clls this fct stble growth. The lower bounds for U usully contrdict the upper bound for U from the liner form estimtes. Stble growth cn be seen considering symptotic expnsions of the u i resulting from 7, but t the time of this writing, we cnnot give ny condition on the fmiliy which gurnties stble growth for n 4; for the cse n = see Thoms [0].. Liner forms in logrithms We give brief survey on some lower bounds for liner forms in logrithms of lgebric numbers which re currently used for solving diophntine equtions. For n lgebric number γ with miniml polynomil d i=0 ix i nd conjugtes γ = γ,..., γ d, the bsolute logrithmic Weil height of γ is defined by hγ := d log [ d d i= mx ], γ i. In generl situtions, one cn use the following estimte of Bker nd Wüstholz []: Theorem. Bker-Wüstholz. Let γ,..., γ n be lgebric numbers, not 0 or, K = Qγ,..., γ n nd d the degree [K : Q]. For i =,..., n let h i mx hγ i, logγ i,. d d Let b,..., b n Z, Λ = b log γ b n log γ n 0 nd B mx b j. Then we hve log Λ > Cn, dh h n log B, where Cn, d = 8n +!n n+ d n+ lognd. In mny concrete fmilies, it is possible to reduce the number of logrithms in the liner form nd to use estimtes for liner forms in few logrithms. Voutier [] considers three logrithms: Theorem. Voutier. Let γ, γ nd γ be positive lgebric numbers nd put D := [Qγ, γ, γ : Q]. Let b, b nd b be integers with b 0 nd let h, h, h, B nd E > be rel numbers which stisfy log E h i mx i, D, hγ i, E log γ i D { B mx, E /D, log E D b h + b h b + b } h h nd E < 4.6 D. If log γ, log γ, nd log γ re linerly independent over Q, then log b log γ + b log γ + b log γ >.4 06 D 5 log B log 4 E h h h. 5

6 Lurent, Mignotte nd Nesterenko [4] settle the cse of two logrithms: Theorem. Lurent-Mignotte-Nesterenko. Let γ nd γ be multiplictively independent nd positive lgebric numbers, b nd b Z nd Λ = b log γ b log γ. Let D := [Qγ, γ : Q], for i =, let { h i mx hγ i, log γ i D, } D nd If Λ 0, then we hve b b D h + b D h. { log Λ 4.4 D mx 4 log b + 0.4, D, } h h. The following very deep conjecture is due to Lng nd Wldschmidt cf. Lng []: Conjecture.4. Let K be n lgebric number field of degree m, β,..., β k K nd b,..., b k Z. Let B,..., B k, B R be rel numbers such tht log B i hβ i, i =,..., k nd B mx{ b,..., b k, e}. Then there exists constnt ck, m > 0 such tht log b log β + + b k log β k > ck, mlog B + + log B k log B, provided tht b log β + + b k log β k 0. Assuming this conjecture, Hlter-Koch, Lettl, Pethő nd Tichy [] could prove: Theorem.5. Let n, = 0,,..., n be distinct integers nd n = n integrl prmeter. Let α = α be zero of P x = n i= x i d with d = ± nd suppose tht the index I of α,..., α n in O, the group of units of O := Z[α], is bounded by constnt J = J,..., n, n for every from some subset Ω Z. Assume further tht the Lng-Wldschmidt conjecture is true. Then for ll but finitely mny vlues Ω the diophntine eqution n x i y dy n = ± i= hs only trivil solutions, except when n = nd =, or when n = 4 nd, {,, ±, ±}, in which cses it hs exctly one more solution for every vlue of. If Qα is primitive over Q especilly if n is prime then there exists bound J = J,..., n, n for the index I by lower bounds for the regultor of O cf. Pohst nd Zssenhus [8], chpter 5.6, 6.. Applying the theory of Hilbertin fields nd results on thin sets, primitivity is proved for lmost ll choices in the sense of density of the prmeters, cf. []. 6

7 A qurtic Fmily of Thue equtions In [7], Pethő nd Tichy considered the two prmetric fmily of Thue equtions F,b X, Y := XX Y X Y X by Y 4 = ±. They proved the following theorem: Theorem.. Assume tht 0 08 < + < b Then hs only the trivil solutions F,b ±, 0 =, + log 4. F,b 0, ± = F,b ±, ± = F,b ±, ± = F,b ±b, ± =. The cse b = + ws not covered by tht pper, becuse its Glois group is different from the generl cse. In the reminder of this pper, we will investigte this cse nd we find ll solutions for ll integers. We will prove: Theorem.. Let be n integer. Then the diophntine eqution F X, Y := XX Y X Y X + Y Y 4 = ± 4 only hs the trivil solutions F ±, 0 =, F 0, ± = F ±, ± = F ±, ± = F ± +, ± =. Let < 0 be n integer nd put A =. Then A 0 nd we hve F X, Y = XX Y X + AY X + A Y Y 4 = ZZ Y Z AY Z A + Y Y 4 = F A Z, Y, with Z = X + AY. Hence it is enough to solve 4 for non-negtive vlues of the prmeter. In [7] it ws proved tht ll solutions x, y Z of 4 with y re exctly the solutions listed in Theorem... Properties of the qurtic number field We put f X := F X, = XX X X + nd we will investigte some properties of the number field Qα, where α is root of f. It is esy to observe tht f is irreducible for 0, the cse = 0 yields precisely the solutions of Theorem. nd will not be considered below. If, ll conjugtes of α re rel, we need shrper pproximtions for the roots of f thn those estblished in [7], Lemm.. Lemm.. Let 7 nd α := α < α < α < α 4 be the zeros of f. Then the following estimtes hold: < α < < α < < α < < α4 <

8 Proof. These inequlities cn esily be verified by considering the sign of f t the given bounds. Sometimes, pproximtions of higher order will be needed; they cn be obtined performing two or three symbolic Newton steps strting t 0,,, + respectively, clculting n symptotic expnsion by Mple nd verifying s in the proof bove. By [7], Theorem., we know tht the Glois group of f is isomorphic to the dihedrl group D 8. Indeed, we hve α 4 = α + + nd α = α + +, since f X ++ = fx. Therefore we hve GlE/Q = 4, 4, where E is the splitting field of f. Moreover, we hve K := Qα = Qα, α 4, hence GlE/K =. Thus there exists exctly one qudrtic subfield of K, sy Qε, nd this subfield is invrint under 4,. This leds to ε = α α 4. In the sequel, we will work in the order O := Z[α]. First we investigte the structure of its unit group O. The corresponding prt of [7] cnnot be used, becuse it depends on the fct tht Qα is primitive over Q in tht sitution. However, the structure is the sme: Theorem.4. Let 0 be n integer, α be root of f = XX X X nd O := Z[α]. If, we hve for the remining vlues of, we hve O =, α, α, α, = : O =, α, α = : O =, α, αα. Proof. We will first discuss the cse 49. To prove tht α, α nd α re independent units, consider the regultor log α log α log α R α := log α log α log α log α log α log α. By Lemm., we obtin for 5 4 log < R α < 4 log +. Thus R α 0 nd the units re independent. We will use the following result of Mhler [9]: Proposition.5 Mhler. Let γ be n lgebric integer of degree d with conjugtes γ = γ,..., γ d nd Mγ := d k= { mx, γ k }. Then discrz[γ] dd Mγ d. Since discr f = nd discr f discr γ, Mhler s result implies Mγ 4/ 4 / 5 8

9 for ll γ O. To prove tht the units re independent, we will first consider nother system of independent units. We will prove tht these units generte O using [8], chpter 5, Theorem 7.. First, we prove tht η := αα > 0 cn be extended to system of fundmentl units. To prove this, we hve to show tht there is no γ O nd no n such tht η = γ n. By 5 nd Lemm. we hve 4/ /n Mγ = Mη /n / 00, 6 00 which is contrdiction for 6. Next, let η := αα > 0. We prove tht η, η cn be extended to system of fundmentl units of O. We hve to prove tht γ n = η k η hs no solution with γ O, n nd k n/. For n 44, we cn rgue s in 6 nd we get contrdiction for 48, since Mη k η Mη k Mη. For 4 n 4, we explicitely bound Mη k η for ll possible choices of k by Lemm. nd we find contrdiction for 4. For n = nd k = 0, we find tht ± η ± η ± η + is no integer for ll choices of the signs, which is impossible since η is n lgebric integer; the cse k = cn be excluded since η < 0. If n =, we find k = 0 d 4d = 8 + ϑ / ϑ [7/8, 8/8] k = d d + d = 4 + ϑ / k = d d + d = ϑ / ϑ [ 4/54, /] ϑ [ /, 55/7], where d i re the symmetric functions in γ i, i. e. d := 4 i= γi nd d := i<j 4 γi γ j, hence d i Z, which is contrdiction for 6. To finish the proof of the cse 49, we use n ide of Lettl nd Pethő [6]. Assume tht, η, η nd some η O generte O. Consider N : O ε ; γ NK/Qε γ = γ γ 4. We see tht N O ε by Pethő [5], Lemm.. η nd η were chosen such tht N η = N η =. Put α = ±η k ηη l m, then we see hence m = ± nd we hve N η m = N η k η l η m = N ±α = ε, O =, η, η, η =, η, η, α =, α, α, α. Next, we consider 48. We used Pri cf. Cohen [8] to compute the regultor R of K for every, we clculted the regultor R α explicitely nd got where M = for ll except I := [ O :, α, α, α ] = R α R O R α R = M, M

10 In these cses we explicitely solved γ n = α k α k α k for ll k, k, k < n/ nd n M. We did not find ny solution γ O with gcdn, k, k, k =. Hence, I = in these cses. The lst step took bout 6 minutes on Pentium 00 running Linux. The cse follows from the positive cse considering f α = 0. The remining cses {, } cn be proved using Knt [9].. Approximtion properties of the solutions Let x, y Z be solution of 4, y. As in, we define the type j of x, y such tht α j x { y = min α k x } y, k 4. By F X + + Y, Y = F X, Y nd Lemm. we see tht if x, y is solution of 4 of type or 4, then x + + y, y is solution of type or, respectively. Thus in order to prove Theorem., we hve to show tht there exists no solution x, y of type or with y. Since we hve F x, y = N Qα/Q x αy =, Theorem.4 yields where η = α, η = α nd η = α. x αy = ±η u ηu ηu, 7. Upper bounds for liner form in logrithms We shll now derive upper bounds for the liner form Λ pqj := u log η p + u log η p + u log η p + log α j α q α j α p, 8 η q η q η q where p nd q will be chosen ccording to the type j of the solution x, y. Furthermore, we will investigte reltions between the u i. Lemm.6. Let 00 nd x, y be solution of 4 with y of type j. The following estimtes hold, ccording to the vlue of j: j = : Let U := u nd V := u u. Then we hve log U + V < U nd U > log. Putting p = nd q =, we hve log Λ < 8 U log + log4.54/. 9 j = : Let U := u u. Then we hve u < U, 7 5 log U + u < U nd U > log. Putting p = nd q = 4 in this cse, we get log Λ 4 < 8 U log + 0 U 9 + log4.54/. 0 Proof. j = : By Lemm., we see α = nd α+ =, thus the continued frction expnsion of α strts with,, α [ ] where + < α < +.. 0

11 Since x/y is principl convergent of α by 4, we hve y. Then 4 yields s in [7], 4.8 α α ν 8 0 < β ν y < α α ν Tking logrithms of the conjugtes of 7, we obtin the following system of liner equtions in the u i : log β 4 = u log + u log + u log log β = u log η + u log η + u log η log β 4 β β 4 = u log η + u log η + u log η By nd Lemm., we hve good estimtes for log β ν /β 4 in terms of. Solving this system by Crmer s rule, we obtin Ru = 6 log log + ϑ log β log + ϑ Ru = log log + ϑ log β 4 log log + + ϑ Ru = log log + ϑ log β 4 log log ϑ 4. where R is the determinnt of the system mtrix, R = 4 log + 5 log nd the ϑ lie in the following intervls: log + ϑ 0, ϑ 0 ϑ ϑ ϑ ϑ ϑ ϑ [ 0., 0.0] [ 5, ] [, 0] [, ] [, ] [, ] [ 6, ]. By nd we hve log β 4 > log. We hve Ru > 5 log log β 4 log 0 > 0 Ru + u 5u > log 4 log β 4 log 4 > 0 R u log u + u 5u > log log β 4 > 0, hence u > log u + u 5u log. By we hve u 4 log Ru 6 log log β log,

12 hence log β 4 log u log. 4 We hve R U + V > log β log 4 5 log > 0 R U log U + V > 5 log log β 4 > 0, which implies tht U > log U + V. Finlly, using 7, Siegel s identity, Lemm. nd, we get Λ = log α α α α β β. β 4 β β 4 β β 4 4. Together with 4, we obtin the requested bound for Λ. [ ] j = : The continued frction expnsion of α strts with, α where < α < 0.9, which yields y This leds to α α ν 4 0 < β ν y < α α ν Tking logrithms of the conjugtes of 7, we obtin the following system of liner equtions in the u i : log β 4 = u log + u log + u log log β = u log η + u log η + u log η log β 4 β β 4 = u log η + u log η Solving this system by Crmer s rule, we obtin where + u log η Ru = log + ϑ log β 4 log log + ϑ Ru = 6 log log + ϑ log β 4 4 log + ϑ Ru = log log + ϑ log β 4 log log 4 + ϑ 4. R = 4 log 5 log nd the ϑ lie in the following intervls: + log + ϑ 0, 7

13 ϑ 0 ϑ ϑ ϑ ϑ ϑ ϑ [0, 0.] [, ] [, ] [, ] [, ] [, ] [ 6, ]. Consider RU < 5 log log β 4 log < 0 RU + u < 4 log β log 4 < 0 R 5U 7 log U + u < log + 4 log log β 4 9 log + 8 log 4 < 0, nd we get 7 log U + u < 5U, hence U 7/5 log. We hve RU > 6 log 5 log log β 4, hence 7 implies log β 4 > log 5 U 9 log. 8 We derive Ru + u + U < log log β 4 log + 5 < 0 R U log u + u + U < log log β 4 < 0, which implies tht U > log u + u + U log. Finlly, using 7, Siegel s identity, Lemm. nd 6, we get Λ 4 = log α α 4 β α α β 4. β 4 β 4 β β β 4 4. Together with 8, we obtin the requested bound for Λ 4..4 Lower bounds for liner form in logrithms Lemm.7. Let 00, x, y solution of 4 of type j with y. Then the following estimtes hold ccording to j: j = : where log Λ > l + U log log Λ > l log 0.8 log U 9, 0 0 l := 4 94.log. log 00 8 log l :=.7 0 log.0 log.

14 j = : where log Λ 4 > l + 5 log U log Λ 4 > l log 7 U log, 0 l := 99 9.log.5 log 00 log 0 l := log 00 log. Proof. j = : We rewrite Λ in the following wy: η η Λ = u log η η + u η η u log η η η + u u + u log + log α α α α Since η = α + + = η nd η = η, the second term vnishes nd we re left with η Λ = U log γ + U + V log γ + log γ = U log γ + log γ γ U+V, 4 where η η γ := η η η η 4, γ := η η, γ := α α α α. We hve hγ 0 4 log 00 8, hγ log, hγ log, log γ, log γ log, log γ log. Applying Theorem. with h := 0 4 log 00 8, h := log + U, b := log 4, we get 9. Putting h := 0 4 log 00 8, h := h := log, E := 8 nd pplying Theorem., we obtin 0. j = : We rewrite Λ 4 in the following wy: η Λ 4 = u log η η4 + u η u log + u U log η + log α α 4 α α. 4 η

15 Since = α + + = η nd = η, the second term vnishes nd we re left with [ α α 4 Λ 4 = log α 4 u + ] +U α α α + α α 4 + U log α 4 α +. Proceeding s bove, we get the estimtes..5 Lrge solutions We compre the upper nd lower bounds for the liner forms given in Lemm.6 nd in Lemm.7: If j =, we get from 9 nd 9 8 l + log4.5 4/ U log l. log If 890, the right hnd side is positive, so we cn insert the lower bound for U from Lemm.6, which yields 8 l + log4.5 4/ log log l. log This is contrdiction for 89. So there exists no solution x, y of type j = nd y for greter thn this bound. For j =, we get in exctly the sme wy contrdiction for Smll solutions To find ll solutions with 00 89, we proceed s in Mignotte, Pethő nd Roth []. First we estblish n explicit upper bound for U in both cses: Lemm.8. Let x, y be solution of 4 of type j with y. j = : Let Then we hve U j = : Let Then we hve U Proof. If j =, we obtin from 0 nd 9 8 U log l + log 0.8 log U. 5 Since U > log, we cn use log x 5 x nd we get U Inserting tht on the right hnd side of 5, we get U nd repeting this process, we get the estimte of the lemm. The cse j = cn be treted in the sme wy. Lemm.9. Let δ, δ, M R, A nd B integers nd A + Bδ + δ < M. 6 Furthermore, let Q N, δ, δ Q with δ i δ i < Q for i =, nd p/q principl convergent of δ with q Q. Then we hve where denotes the distnce to the nerest integer. q q δ Q M + + B, 7 5

16 Proof. Multiplying 6 by q, we hve q δ + qδ δ + qa Bp q δ + Bp Bq δ δ QM, hence nd the ssertion follows. q q δ Q M + q δ δ + B + B q δ δ Let j =. By nd 9 we hve 6 with A := U, B := U + V, M = 0 900, δ := log γ log γ, δ := log γ log γ. We choose Q depending on the vlue of : 00 < Q = < Q = < 89 Q = 0 6 For ech, we compute rtionl pproximtions δ i of δ i nd convergents p/q of δ with q Q. For most vlues of, we find such convergent with q q δ > log, nd this is contrdiction to 7 by Lemm.6 nd Lemm.8. For the remining vlues of, we repet this rgument with Q = 0 0 nd get the corresponding contrdiction. The cse j = is treted in exctly the sme wy, we only give the vlues of Q tht we hve used: 00 < Q = < Q = 0 5. Hence there re no nontrivil solutions if 00. For the cse 99, we used progrm of Hnrot solving Thue equtions following the lgorithm of Bilu nd Hnrot [4], where the fundmentl units which re known by Theorem.4 cn be explicitly given. This took 50 seconds on Pentium 00 nd only gve the trivil solutions known from Theorem.. Thus Theorem. is proved. The computtions were performed on DEC Alph worksttion nd on Pentium 00 running Linux of TU Grz. We hve used MAPLE V in the forml computtions, Pri s librry mode for the exclusion of the existence of smll solutions. We hve done this prt of the clcultions twice, first we only used rtionl numbers in the continued frction procedure this took dys for j = nd 8 dys for j = on the DEC Alph, then we used high precision rel numbers 8 hours for j = nd 7 hours for j = on the Pentium..7 Avoiding Voutier s estimte In the numericl clcultions we hve used result of Voutier on liner forms in three logrithms, which is not published yet. In this section, we pply the generl theorem of Bker nd Wüstholz [] Theorem. insted of Voutier [] Theorem.. Sections. to. do not depend on Theorem.. In Lemm.7, estimtes 0 nd re ffected. We replce them by 6

17 Lemm.0. Let 00, x, y be solution of 4 with y nd Λ pqj be the liner form defined in 8 with p, q, j defined s in Lemm.6. Then the following estimte holds: log Λ pqj > l log U 8 where l =. 0 7 log. Proof. We prove this ssertion depending on the type j of the solution x, y. j = : We pply Theorem. on the liner form Λ = U log γ + V U log γ + log γ, where γ, γ, γ hve been defined in the proof of Lemm.7, using h := log, h := log, h := log nd obtin 8. j=: By the considertions in the proof of Lemm.7 we obtin where Λ 4 = u + U log γ U log γ + log γ, γ := α α 4, γ := We pply Theorem. with nd obtin 8. α α α 4 α, γ 4 := α α 4 α α. h := 4. 4 log, h := log, h = log Section.5 is independent from Voutier s estimte, hence we still hve 89 if j = or if j =. However, in Section.6 we hve to substitute Lemm.8 by the following: Lemm.. Let x, y be solution of 4 of type j with y. If 00 j where = 89 nd = , we hve U 0. Proof. j = : Combining 9 nd 8, we obtin 8 U log l + log U. 9 Since by Lemm.6 we hve U > log > for 00, we cn use the estimte log U U /4. Hence 9 implies U 0 6. Inserting this bound in the right side of 9, we get U 0. j = : The combintion of 0 nd 8 implies which yields U 0 s in the cse j =..64U log l + log U, 40 7

18 Of course, Lemm.9 holds unchnged. For the finl rgument of Section.6 we observe tht 6 holds with A := U, B := U + V, M = 0 900, δ := log γ log γ, δ := log γ log γ if j = nd with A := U, B := u + U, M = 0 900, δ := log γ log γ, δ := log γ log γ if j =. In both cses, we choose Q = 0 0, compute rtionl pproximtions δ i of δ i for i =, nd convergents p/q of δ with q Q. For 00 j, we find such convergent with q q δ > 0 log, which is contrdiction to 7 by Lemm.6 nd Lemm.8. The numericl computtions re bsed on the following error estimte: Lemm.. Let 00 j be n integer, 0 < ε < 00 nd αk Q with α k α k < ε for k 4. Then we hve δ i δ i < Ej ε, i =, where E = nd E = Proof. Simple error estimtes. Hence the roots of f hve been clculted with precision of 64 nd 80 bytes 54 nd 9 decimls in cses j = nd j =, respectively, which gve the required precision Q for δ i. The computtions hve been performed using Pri in librry mode on network of three PCs running Linux nd Alph Worksttion running Digitl Unix, which took totl time of pproximtely two weeks. Acknowledgement: We re grteful to Mr. Guillume Hnrot for dpting his progrm to our purposes. References [] A. Bker. Contribution to the theory of Diophntine equtions I. On the representtion of integers by binry forms. Philos. Trns. Roy. Soc. London Ser. A, 6:7 9, 968. [] A. Bker nd H. Dvenport. The equtions x = y nd 8x 7 = z. Qurt. J. Mth. Oxford, 0:9 7, 969. [] A. Bker nd G. Wüstholz. Logrithmic forms nd group vrieties. J. reine ngew. Mth., 44:9 6, 99. [4] Yu. Bilu nd G. Hnrot. Solving Thue Equtions of High Degree. J. Number Theory, 60:7 9, 996. [5] E. Bombieri nd W. M. Schmidt. On Thue s eqution. Invent. Mth., 88:69 8, 987. [6] Y. Bugeud nd K. Győry. Bounds for the solutions of Thue-Mhler equtions nd norm form equtions. Act Arith., 74:7 9,

19 [7] J. H. Chen nd P. M. Voutier. Complete solution of the Diophntine Eqution X + = dy 4 nd Relted Fmily of Qurtic Thue Equtions. J. Number Theory, 6:7 99, 997. [8] H. Cohen. A Course in Computtionl Algebric Number Theory, volume 8 of Grdute Texts in Mthemtics. Springer, Berlin etc., third edition, 996. [9] M. Dberkow, C. Fieker, J. Klüners, M. E. Pohst, K. Roegner, nd K. Wildnger. KANT V4. J. Symbolic Comput., 4:67 8, 997. [0] I. Gál. On the resolution of some diophntine equtions. In A. Pethő, M. Pohst, H. C. Willims, nd H. G. Zimmer, editors, Computtionl Number Theory, pges De Gruyter, Berlin New York, 99. [] F. Hlter-Koch, G. Lettl, A. Pethő, nd R. F. Tichy. Thue equtions ssocited with Ankeny-Bruer-Chowl Number Fields. To pper in J. London Mth. Soc. [] C. Heuberger. On Fmily of Quintic Thue Equtions. J. Symbolic Comput., 6:7 85, 998. [] S. Lng. Elliptic Curves: Diophntine Anlysis, volume of Grundlehren der Mthemtischen Wissenschften. Springer, Berlin New York, 978. [4] M. Lurent, M. Mignotte, nd Yu. Nesterenko. Formes linéires en deux logrithmes et déterminnts d interpoltion. J. Number Theory, 55:85, 995. [5] E. Lee. Studies on Diophntine equtions. PhD thesis, Cmbridge University, 99. [6] G. Lettl nd A. Pethő. Complete Solution of Fmily of Qurtic Thue Equtions. Abh. Mth. Sem. Univ. Hmburg, 65:65 8, 995. [7] G. Lettl, A. Pethő, nd P. Voutier. Simple fmilies of Thue inequlities. To pper in Trns. Amer. Mth. Soc. [8] G. Lettl, A. Pethő, nd P. Voutier. On the rithmetic of simplest sextic fields nd relted Thue equtions. In K. Győry, A. Pethő, nd V. T. Sós, editors, Number Theory, Diophntine, Computtionl nd Algebric Aspects. Proceedings of the Interntionl Conference held in Eger, Hungry, July 9 August, 996. W. de Gruyter Publ. Co., 998. [9] K. Mhler. An inequlity for the discriminnt of polynomil. Michign Mth. J., :57 6, 964. [0] M. Mignotte. Pethő s cubics. Preprint. [] M. Mignotte. Verifiction of Conjecture of E. Thoms. J. Number Theory, 44:7 77, 99. [] M. Mignotte, A. Pethő, nd R. Roth. Complete solutions of qurtic Thue nd index form equtions. Mth. Comp., 65:4 54, 996. [] M. Mignotte, A. Pethő, nd F. Lemmermeyer. On the fmily of Thue equtions x n x y n + xy y = k. Act Arith., 76:45 69, 996. [4] M. Mignotte nd N. Tznkis. On fmily of cubics. J. Number Theory, 9:4 49, 99. 9

20 [5] A. Pethő. Complete solutions to fmilies of qurtic Thue equtions. Mth. Comp., 57: , 99. [6] A. Pethő nd R. Schulenberg. Effektives Lösen von Thue Gleichungen. Publ. Mth. Debrecen, 4:89 96, 987. [7] A. Pethő nd R. F. Tichy. On Two-prmetric Qurtic Fmilies of Diophntine Problems. J. Symbolic Comput., 6:5 7, 998. [8] M. Pohst nd H. Zssenhus. Algorithmic lgebric number theory. Cmbridge University Press, Cmbridge etc., 989. [9] E. Thoms. Complete Solutions to Fmily of Cubic Diophntine Equtions. J. Number Theory, 4:5 50, 990. [0] E. Thoms. Solutions to Certin Fmilies of Thue Equtions. J. Number Theory, 4:9 69, 99. [] A. Thue. Über Annäherungswerte lgebrischer Zhlen. J. reine ngew. Mth., 5:84 05, 909. [] N. Tznkis nd B. M. M. de Weger. On the prcticl solution of the Thue eqution. J. Number Theory, :99, 989. [] P. Voutier. Liner forms in three logrithms. Preprint. [4] I. Wkbyshi. On Fmily of Qurtic Thue Inequlities I. J. Number Theory, 66:70 84, 997. Clemens Heuberger Institut für Mthemtik Technische Universität Grz Steyrergsse 0 A-800 Grz Austri Attil Pethő Mthemticl Institute Kossuth Ljos University P.O. Box H-400 Debrecen Hungry Robert F. Tichy Institut für Mthemtik Technische Universität Grz Steyrergsse 0 A-800 Grz Austri 0