On a Family of Quintic Thue Equations

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1 J. Symbolic Computtion 998) 6, 7 85 Article No. sy98006 On Fmily of Quintic Thue Equtions CLEMENS HEUBERGER Institut für Mthemtik, Technische Universität Grz, Steyrergsse 0, A-800 Grz, Österreich We consider the prmetrized quintic fmily of Thue equtions XX Y )X Y ) Y 5 = ± Z nd prove tht it only hs trivil solutions for using recent estimte for liner forms in three logrithms of lgebric numbers by P. Voutier. c 998 Acdemic Press. Introduction Let F Z[X, Y ] be n irreducible form of degree nd m nonzero integer. The diophntine eqution F X, Y )=m.) is clled Thue eqution in honour of Thue 909), who proved tht.) only hs finite number of solutions x, y) Z. His result ws consequence of his theorem on diophntine pproximtion, which is not effective. Bker 968) gve n effective upper bound for mx x, y ) bsed on his studies on liner forms in logrithms of lgebric numbers; pplying numericl reduction method due to Bker nd Dvenport 969), it is possible to clculte ll solutions of single Thue eqution with computer, see Pethő nd Schulenberg 987), Tznkis nd de Weger 989), Bilu nd Hnrot 996)). Recently, severl prmetrized fmilies of Thue equtions hve been investigted; Thoms 990) who ws the first to consider prmetric fmilies Mignotte et l. 996b), Lee 99), Mignotte nd Tznkis 99), nd Thoms 99) considered cubic fmilies. Qurtic fmilies hve been solved by Pethő 99), Mignotte et l. 996), Lettl nd Pethő 995), Chen nd Voutier 997), nd Pethő nd Tichy 997); in Lettl et l. 997, b), sextic fmily hs been completely solved. Hlter-Koch et l. 997) considered very generl fmily of rbitrry degree, ssuming very deep conjecture of Lng nd Wldschmidt see Lng 978)). In this pper we consider the fmily of quintic Thue equtions We will prove: F X, Y ):=XX Y )X Y ) Y 5 = u = ±..) Theorem.. Let Then the only solutions of.) re F ±, 0) = ± nd F 0, ±) = F, ±) = F, ±) = F, ±) = F, ±) = /98/ $0.00/0 c 998 Acdemic Press

2 7 C. Heuberger Using Ksh see Dberkow et l. 997)), Theorem. hs been verified for 00, which took bout week on DEC Alph with 75 MHz, we conjecture tht the theorem is vlid for ll Z. For some comments on the size of the constnt in the Theorem see the remrk t the end of the pper. Put f X) :=F X, ) = XX )X ) nd denote its roots by α = α ),...,α 5). For solution x, y) of.) we hve 5 F x, y) = x α ν) y)=n Qα)/Q x αy) =±..) ν= This mens tht x αy is unit in the order O := Z[α]. To prove Theorem., we will first investigte the structure of O, the unit group of O, then we will define liner forms in logrithms, derive upper bounds for them contrdicting to the lower bounds, given by theorem of Voutier 997), for lrge vlues of. Most of the clcultions involve hevy mnipultions with symptotic pproximtions, they hve been crried out by some kind of symbolic intervl rithmetic using the computer lgebr system Mple.. Elementry Observtions Since F = F, it suffices to consider nonnegtive vlues of. ByF X, Y )= F X, Y ), we only hve to consider solutions x, y) with y 0. If y = 0, then x =;ify = we hve either xx )x )=0, which hs the solutions indicted in Theorem., or xx )x )=, which leds to x )xx +), contrdiction. Therefore, we cn suppose y. Throughout the pper, we will need estimtes for the roots α ν), ν =,...,5. Lemm.. Let. Then f hs five rel roots α ) >α ) >α ) >α ) >α 5) stisfying the following estimtes for : α) α) α) α) α5) Proof. These inequlities cn esily be verified by considering the sign of f t the points ,...

3 On Fmily of Quintic Thue Equtions 75 Sometimes, pproximtions of higher order will be needed. These cn be obtined by performing two or three symbolic Newton steps strting t,, 0,, respectively, clculting n symptotic expnsion by Mple nd verifying s in the proof bove. Clerly, there is no liner fctor dividing f, indeed, it is elementry to show tht f is irreducible for ll vlues Z.. Algebric Properties of the Number Field Since solution of.) corresponds to unit in O, we hve to investigte the unit group of this order. Theorem.. Let, α be root of f nd O = Z[α]. Then O =,α,α,α+,α. Proof. Since f α) = 0, we see tht α, α,α+,α re units in O. We will derive n upper bound for the index of,α,α,α+,α in the unit group of O by estimting the regultors of the two groups. The discriminnt D O = Df ) is given by Df )= > 5.99 for 8. By Pohst nd Zssenhus 989, chpter V, 6.)), the regultor of O cn be estimted s follows: [ log D O /6) R O 5 6 ) ] 0 log D O /6) log ) Let R α be the regultor of,α,α,α+,α : log α ) log α ) log α ) + log α ) R α = log α ) log α ) log α ) + log α ) log α ) log α ) log α ) + log α )..) log α ) log α ) log α ) + log α ) Appliction of Lemm. yields for 0: 0 log <R α < 0 log log + log log Hence R α > 0 nd α, α,α+,α re independent units. Now, we cn bound the index I := [O :<,α,α,α+,α >]by I = R α R O < 0 log log + log log log 0.00 log +0.6 log 0 log <.) for 686. Hence I {,, } in this cse. In order to exclude the cses I = nd I =, we need the following result of Mhler 96): Proposition.. Let γ be n lgebric integer of degree d with conjugtes γ = γ ),...,γ d) nd d { } Mγ) := mx, γ k). k=.

4 76 C. Heuberger Then DZ[γ] dd Mγ) d ). In our cse, this yields for 8 nd γ O\Z ) / Mγ) 7/..) Assume now I =. Then there exists n ε O stisfying 5 5 ε = α i α ) j α +) k α ) l.) with integers i, j, k, l, becuse we cn ssume ε to be positive. Furthermore we cn ssume i, j, k nd 0 l. As K = Qα) is quintic number field, it is primitive, nd ε is quintic lgebric integer. Let 5 5 hx) = X ε i) )= d j X j i= be its miniml polynomil with d j Z. Appliction of Lemm. yields j=0 Mα) <.0, Mα ) <.0, Mα +)<.0. First ssume tht l = 0 nd tht t most two of the three numbers i, j, k re nonzero. By.),.), Mγ n )=Mγ) n nd Mγ γ ) Mγ )Mγ )wehve ) / / Mε) ) /, 5 5 which is contrdiction for 8. Consider now the cse i = j = k = nd l = 0. Then Mhler s estimte does not help us, but using pproximtions of α ν) of order 0 nd symptotic expnsions of third roots, we obtin for / <d < + +. /. Therefore d cnnot be n integer, contrdiction. As third cse, consider i, j, k, l) =,, 0, ), which is equivlent to i, j, k, l) =,, 0, ). In this cse, we obtin / < ) d + + ) d d +d < /. The expression in the middle should be n integer, but this is impossible for 097. Actully, ll possible cses occurring for the exponents i, j, k, l cn be delt with by using one of the bove three types of rguments, if 765 ; for complete list see Heuberger 997). The cse I = cn be treted in the sme wy, here we hve to be creful of the signs of the ε ν), complete list cn be received from the uthor. For the index bound.) hs been computed explicitly nd the

5 eqution ε n = α i α ) j α +) k α ) l On Fmily of Quintic Thue Equtions 77 hs been checked for ll possible tuplets n, i, j, k, l) using the computtionl number theory system PARI. In ll cses the only solution ws i = j = k = l = 0. This verifiction took bout week on Pentium 00 computer. We will lso work in the splitting field of f, so we investigte its Glois group. Theorem.. For Z, we hve Glf )=S 5. Proof. According to Cohen 996), Algorithm 6..9, we hve to check tht R = ) X F α σ)),...,α σ5)) ) σ H with nd F x,x,x,x,x 5 )=x x x 5 + x x )+x x x + x x 5 )+x x x 5 + x x ) +x x x + x x 5 )+x 5 x x + x x ) H = {id, ), ), ), 5), 5)} does not hve n integrl root. But this cn immeditely be checked by clculting bounds for ε σ := F α σ)),...,α σ5)) ) with σ H pplying Lemm... Approximtion Properties of the Solutions Let x, y) Z be solution of.) with y. For ν =,...,5, we define β ν) := x α ν) y, which re units by.). Writing η ν) := α ν), η ν) := α ν), η ν) := α ν) +, η ν) := α ν) nd pplying Theorem. we hve β ν) = ± with u,...,u Z. We define η ν) ) u η ν) ) u η ν) ) u U := mx{ u,..., u }. η ν) ) u.) We will now use stndrd mteril cf. Bilu nd Hnrot 996)) to derive symptotic expressions for the β ν). By.) we see tht 5 x y αν) = y 5, ν= hence x/y is n pproximtion to some α ν). To record this, we define the index j by β j) = min β ν) ν {,...,5} nd sy tht x, y) is solution of type j. Then the following lemm holds:

6 78 C. Heuberger Lemm.. hve Let nd x, y) be solution of.) of type j with y. Then we x α j) 6 y y f α j) ) y,.) hence x/y is principl convergent of α j), nd for ν j x α ν) α ν) α j) y y..) Proof. We hve y α ν) α j) x α ν) y + x α j) y x α ν) y, nd.) is proved. Furthermore,.) gives f α j) ) = α ν) α j) 6 y x α ν) 6 y = y x α j) y. ν j ν j Appliction of Lemm. yields for + 6 f α ) ) + 8.) + f α ) ) + 6.5) f α ) ).6) f α ) ).7) 8 f α 5) ) 6,.8) hence f α j) ) /, nd the lemm is proved, since we ssume y. 5. A Liner Form in Logrithms of Algebric Numbers For pirwise distinct l, p, q {,...,5} Siegel s identity holds: x α p) y x α q) y αq) α l) α p) α = x αl) y l) x α q) y αq) α p) 5.) α p) α l) If we choose l = j, the right-hnd side will become smll, nd so by.) nd Lemm. the bsolute vlue of the liner form ν p) Λ p,q,j := u log + u ν p) log + u ν p) log + u ν p) log + log α p) α j) α p) α j) η q) η q) will be very smll. According to the type of x, y), we will now choose p nd q, give n upper bound for this liner form nd investigte reltions between the u i. Lemm 5.. Let 089 nd x, y) be solution of.) of type j with y. The following estimtes hold, ccording to the vlue of j: j {, 5}: U>c j log, u u <U/ log ) nd u u u <U/c j log ), η q) η q)

7 On Fmily of Quintic Thue Equtions 79 where c =nd c 5 =. We hve log Λ,,j < H j U log with H =5nd H 5 = 500/0. j {, }: U> log, u <U/ log ) nd u + u + u <U/c j log ), where c =/ nd c =. We hve log Λ,5,j < 5/)U log. j =: U> log, u <U/ log ) nd u u < U)/ log ). We hve log Λ,5, < 500/0)U log. The symmetric nture of our eqution.) enbled us to collect similr cses, s cn be seen in this lemm. Proof. We prove this lemm only for j =, becuse the proofs of the other cses re nlogous. Lemm. yields α ) = 0 nd α =, hence the continued frction expnsion [ ) of α ) strts with Therefore, we hve 0,,α ) ], where.5+ <α) = α) α < ).5+. α ) = 5 nd hence by Lemm. we obtin y 5 >. 5.) By.),.5) nd 5.) we hve y <β) = x α ) y< 6 y 6 0, hence α ) 6 < x y <α) + 6, nd for ν y α ) α ν) 6 ) <β ν) = x α ν) y<y α ) α ν) + 6 ). 5.) Tking logrithms of the conjugtes of.), we hve the following system of liner equtions in the u i : log β 5) = u log η 5) + u log η 5) η 5) + u + u log η 5) log β ) β 5) = u η ) log η 5) + u η ) log η 5) + u η ) log η 5) + u η ) log η 5) log β ) β 5) = u η ) log η 5) + u η ) log η 5) + u η ) log η 5) + u η ) log η 5) log β ) = u η ) log + u η ) log + u η ) log + u η ) log β 5) η 5) η 5) By 5.) nd Lemm., we hve good estimtes for log β ν) /β 5) in terms of. η 5) η 5)

8 80 C. Heuberger Solving the system by Crmer s rule, we obtin Ru = 0 log log log log log ) log + ϑ log β 5) where +0 log + log log + log log log + ϑ Ru = 0 log + log log + 6 log log ) log + ϑ log β 5) 0 log log log + log log log + ϑ Ru = 0 log + log log + 6 log log ) log + ϑ log β 5) +0 log log log 6 log log log + ϑ ) Ru = 0 log + ϑ log β 5) log log +0 + ϑ, R = R α > 0 log + 8 log log + 8 log log 5.) is the regultor of Z[α ) ] see.)) nd the ϑ ik lie in the following intervls: ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ [, 9] [0, ] [, 9] [, 0] [, 9] [8, 0] [, ] [9, ] By 5.) nd 5.), we hve β 5), hence log β 5) log. This yields Ru > 0, Ru < 0 nd U = u. Furthermore we hve Ru + Ru log log + log ) log β 5) + 0 log log 5 > 0. Since u is n integer, this yields From 5.) we conclude U = u > log u log. 5.5) U0 log + 8 log log + 8 log log ) Ru = Ru 0 log + log log + 6 log log ) log β 5) 0 log log log + log log, nd which together with 5.5) implies log β 5) U log log >log. 5.6) Moreover, we hve Ru + Ru + Ru 50 log ) log β 5)

9 On Fmily of Quintic Thue Equtions 8 +0 log + 0 log + log +50< 0 nd Ru +log Ru + Ru + Ru ) > 0 log + 6 log + 8 log 76) log β 5) +0 log log + 7 log 7 > 0, which yields u + u + u U//) log ). Putting l = j =,p = nd q = 5 in Siegel s identity 5.) nd using.), Lemm. nd Lemm., we obtin ) η ) u ) η ) u ) η ) u ) η ) u α 5) α ) η 5) η 5) η 5) η 5) α ) α ) = x α ) y x α 5) y α5) α ) α ) α ). β ) 6 β 5). y α 5) α ) 00 y, hence we hve Λ,5, <.5 Together with 5.6) this mens β ) β 5) =.5 ) β 5) β ) β ) β ) β 5) 5 <.8 β 5) 5. log Λ,5, < log.8 ) 5 U log 9 log < 5 U log. 6. A Lower Bound for Liner Form in Logrithms We rewrite the liner form Λ p,q,j defined in Lemm 5. in the following wy s liner form in three logrithms: for j {, 5} we write η ) η ) η ) Λ,,j = u log η ) η ) η ) + u η ) log η ) + log α ) α j) η ) η ) u u α ) α j) η ) u +u u, for j {, } Λ,5,j = u log nd for j = Λ,5, = u log η ) η 5) η ) η 5) η ) η 5) + u log η ) η ) η ) η ) + u log η 5) η 5) + log α 5) α j) α ) α j) η ) η 5) η ) η 5) η ) η 5) + log α 5) α ) α ) α ) η ) u +u +u η ) η 5) u u We know n upper bound for this liner form in logrithms by Lemm 5., we will now η ) η 5) η ) η 5) u u.

10 8 C. Heuberger derive lower bound using recent result of Voutier 997). 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 s [ hγ) := d ) ] d log mx, γ i). d i= Proposition 6.. Voutier) Let γ, γ nd γ be positive lgebric numbers nd put D := [Qγ,γ,γ ):Q]. Letb, b nd b be integers with b 0nd let h, h, h, B nd E> be rel numbers which stisfy log E h i mx D,hγ i), E log γ ) i i, 6.) D { B mx,e /D, log E b D + b ) b + b )} 6.) h h h h nd E<.6 D.Iflog γ, log γ, nd log γ re linerly independent over Q, then log b log γ + b log γ + b log γ >. 06 D 5 log B log E Applying this proposition, we obtin h h h. Lemm 6.. Let 089 nd x, y) be solution of.) of type j with y. We define h, h, h, nd B ccording to the vlue of j: j {, 5}: Define h := log, h := 0 log670 ), h := log ) + log6. ) 0 log 0 log, nd B := 0.). j {, }: Define h := 0 log656 ), h := 0 log ), h := nd B := 0.0) or 0.0) if j =or j =, respectively. j =: Define 0 log log6 ), h := 0 log.000 ), h := 6 5 log, h := log6. ) 80 log + log670 ) 80 log nd B := 0.09 ). Furthermore define h := 60 log85 9 ), h := h + Uh, C := log. Then we hve where p nd q re defined in Lemm 5.. log Λ p,q,j > C log Bh h h,

11 On Fmily of Quintic Thue Equtions 8 Proof. We prove the lemm for j =, ll remining cses re nlogous. First, we hve to check the liner independnce of the logrithms. Assume tht they re linerly dependent. Since Gl f = S 5, we cn pply the Glois utomorphisms induced by σ = ) nd σ = ), which yields α 5) α ) α ) α ) = α 5) α ) α ) α ) = α 5) α ) α ) α ), becuse Qα ),...,α 5) ) is totlly rel number field. But this leds to contrdiction. In our cse, D = [ Qα ),α ),α 5) ):Q ] = 60; we choose E := nd estimte the heights using pproximtions of the α ν) of order : h log η ) η 5) η ) η 5) h log η ) η 5) η ) η 5) η 5) η ) η 5) η ) η 5) η ) η 5) η ) 0 log656 ), ) 0 log ) )., nd h nd h stisfy 6.). To estimte the height of α5) α ) which, in generl, is not n lgebric integer, we α ) α ) hve to estimte the leding coefficient 60 of its miniml polynomil. Since resultnt Y f X + Y ),f Y )) = X 5 X Df )) nd [ Qα p) α q) ):Q ] = 0 for p q by Theorem., the miniml polynomil mx) of α p) α q) is mx) =X Df ) nd the miniml polynomil mx) ofα p) α q) ) is Now α5) α ) α ) α ) isrootof resultnt Y Y 0 m X Y mx) =Df )X ) ), my ) = Df ) 0 p q r s nd, therefore, the leding coefficient 60 Df ) 0. Hence we obtin α 5) α ) ) h α ) α ) 60 log85 9 ) log α 5) α ) α ) α ). X αp) α q) ) Z[X], α r) α s)

12 8 C. Heuberger h log h log η ) η 5) η ) η 5) η ) η 5) η ) η 5) ) 0 log6. ) ) 0 log670 ) log5 ), where the lst two lines determined the choice of E, hence by Lemm 5. nd becuse of h = 0 log log056 ) log 0 log6. )+ log 0 log670 ) h stisfies 6.). For our ppliction b, b U nd b =. Hence for i =, wehve b i h + b h i h + h i log, nd 6.) is stisfied with B =0.0). Hence the lemm follows from Proposition Proof of Theorem. With the nottions of the previous sections nd putting j 5 H j 5 5/ 500/0 5/ 500/0 â j j , Lemmt 5. nd 6. yield Hence H j U log >log Λ p,q,j > Ch h h log B. Ch h h log B>U H j log Ch h h log B ) =: U g). 7.) For â j, g) is positive, nd therefore we cn insert the lower bound for U from Lemm 5., nd 7.) cnnot be true for j, hence the ssumption tht solution with y> exists leds to contrdiction nd Theorem. is proved. Remrk. The size of the constnts j depends minly on the constnt C in Lemm 6., which comes from the fctor D 5 in Proposition 6., which is very high becuse of the Glois group S 5.

13 On Fmily of Quintic Thue Equtions 85 Furthermore, we hve to use liner forms in three logrithms, becuse in the cses j =,,, 5, there is no dominting exponent u i, which would enble us to group the liner form into liner form in two logrithms, where the constnts would be fr better. Acknowledgements This reserch ws supported by the Austrin Hungrin Science Coopertion project nd the Austrin Ntionl Bnk Jubiläumsfonds) Nr. 995 References Bker, A. 968). Contribution to the theory of Diophntine equtions I. On the representtion of integers by binry forms. Phil. Trns. Roy. Soc. London Ser. A, 6, 7 9. Bker, A., Dvenport, H. 969). The equtions x =y nd 8x 7=z. Qurt. J. Mth. Oxford, 0, 9 7. Bilu, Y., Hnrot, G. 996). Solving Thue equtions of high degree. J. Number Theory, 60, 7 9. Chen, J. H., Voutier, P. M. 997). Complete solution of the Diophntine eqution X +=dy nd relted fmily of qurtic Thue equtions. J. Number Theory, 6, Cohen, H. 996). A Course in Computtionl Algebric Number Theory, volume 8 of Grdute Texts in Mthemtics. Springer, rd edn. Dberkow, M., Fieker, C., Klüners, J., Pohst, M. E., Roegner, K., Wildnger, K. 997). KANT V. J. Symb. Comput., Hlter-Koch, F., Lettl, G., Pethő, A., Tichy, R. F. 997). Thue equtions ssocited with Ankeny-Bruer- Chowl Number Fields. J. London Mth. Soc. in press) Heuberger, C. 997). Algorithmische Lösung prmetrisierter Thue-Gleichungen. Diplomrbeit, Technische Universität Grz. Lng, S. 978). Elliptic Curves: Diophntine Anlysis, volume of Grundlehren der Mthemtischen Wissenschften. Springer. Lee, E. 99). Studies on Diophntine equtions. PhD thesis, Cmbridge University. Lettl, G., Pethő, A. 995). Complete solution of fmily of qurtic Thue equtions. Abh. Mth. Sem. Univ. Hmburg, 65, Lettl, G., Pethő, A., Voutier, P. 997). On the rithmetic of simplest sextic fields nd relted Thue equtions. In Győry, K., Pethő, A., Sós, V. T., eds, Number Theory, Diophntine, Computtionl nd Algebric Aspects. W. de Gruyter Publ. Co. Lettl, G., Pethő, A., Voutier, P. 997b). Simple fmilies of Thue inequlities. Trns. Am. Mth. Soc. in press). Mhler, K. 96). An inequlity for the discriminnt of polynomil. Michign Mth. J.,, Mignotte, M., Pethő, A., Roth, R. 996). Complete solutions of qurtic Thue nd index form equtions. Mth. Comp., 65, 5. Mignotte, M., Pethő, A., Lemmermeyer, F. 996b). On the fmily of Thue equtions x n )x y n +)xy y = k. Act Arith., 76, Mignotte, M., Tznkis, N. 99). On fmily of cubics. J. Number Theory, 9, 9. Pethő, A. 99). Complete solutions to fmilies of qurtic Thue equtions. Mth. Comp., 57, Pethő, A., Schulenberg, R. 987). Effektives Lösen von Thue Gleichungen. Publ. Mth. Debrecen,, Pethő, A., Tichy, R. F. 997). On two-prmetric qurtic fmilies of diophntine problems. J. Symb. Comput. in press) Pohst, M., Zssenhus, H. 989). Algorithmic Algebric Number Theory. Cmbridge University Press. Thoms, E. 990). Complete solutions to fmily of cubic diophntine equtions. J. Number Theory,, Thoms, E. 99). Solutions to certin fmilies of Thue equtions. J. Number Theory,, Thue, A. 909). Über Annäherungswerte lgebrischer Zhlen. J. reine ngew. Mth., 5, Tznkis, N., de Weger, B. M. M. 989). On the prcticl solution of the Thue eqution. J. Number Theory,, 99. Voutier, P. 997). Liner forms in three logrithms. Preprint. Originlly received 0 June 997 Accepted December 997