ON EQUAL VALUES OF POWER SUMS OF ARITHMETIC PROGRESSIONS

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1 GLASNIK MATEMATIČKI Vol 4767)01), ON EQUAL VALUES OF POWER SUMS OF ARITHMETIC PROGRESSIONS András Bzsó, Dijn Kreso, Florin Luc nd Ákos Pintér University of Debrecen, Hungry, Technische Universität Grz, Austri nd Universidd Ncionl Autónom de México, Mexico Abstrct In this pper, we consider the Diophntine eqution b k ++b) k + +x 1)+b) k = = d l +c+d) l + +cy 1)+d) l, where,b,c,d,k,l re given integers with gcd,b) = gcdc,d) = 1, k l We prove tht, under some resonble ssumptions, the bove eqution hs only finitely mny solutions For positive integer n, let 1 Introduction nd results 11) S k,b n) = bk ++b) k + +n 1)+b) k It is esy to see tht the bove power sum is relted to the Bernoulli polynomils B k x) in the following wy: S,b k k n) = [B k+1 n+ b ] ) B k+1 k +1 1) [ ]) b B k+1 ) B k+1, where the polynomils B k x) is defined by the generting series texptx) expt) 1 = B k x) tk k! k=0 010 Mthemtics Subject Clssifiction 11B68, 11D41 Key words nd phrses Diophntine equtions, exponentil equtions, Bernoulli polynomils 53

2 54 A BAZSÓ, D KRESO, F LUCA AND Á PINTÉR nd B k+1 = B k+1 0) For the properties of Bernoulli polynomils which will be often used in this pper, sometimes without specil reference, we refer to [7, Chpters 1 nd ] We cn extend S,b k for every rel vlue x s 13) S,b k k x) = B k+1 x+ b ) )) b B k+1 k +1 We denote by C[x] the ring of polynomils in the vrible x with complex coefficients A decomposition of polynomil Fx) C[x] is n equlity of the following form which is nontrivil if Fx) = G 1 G x)) G 1 x),g x) C[x]), degg 1 x) > 1 nd degg x) > 1 Two decompositions Fx) = G 1 G x)) nd Fx) = H 1 H x)) re sid to be equivlent if there exists liner polynomil lx) C[x] such tht G 1 x) = H 1 lx)) nd H x) = lg x)) The polynomil Fx) is clled decomposble if it hs t lest one nontrivil decomposition; otherwise it is sid to be indecomposble In recent pper, Bzsó, Pintér nd Srivstv [1]) proved the following theorem bout the decomposition of the polynomil S,b k x) defined bove Theorem 11 The polynomil S,b k x) is indecomposble for even k If k = v 1 is odd, then ny nontrivil decomposition of S,b k x) is equivlent to the following decomposition: 14) S,b k x) = Ŝv x+ b 1 ) ) Proof This is [1, Theorem ] Using Theorem 11 nd the generl finiteness criterion of Bilu nd Tichy [])fordiophntineequtionsoftheformfx) = gy), weprovethe following result Theorem 1 For k < l, the eqution 15) S k,bx) = S l c,dy) hs only finitely mny solutions in integers x nd y Since the finiteness criterion from [] is bsed on the ineffective theorem of Siegel, our Theorem 1 is ineffective We note tht for = c = 1,b = d = 0 our theorem gives the result of Bilu, Brindz, Kirschenhofer, Pintér nd Tichy [3]) Combining result of Brindz [5] with recent theorems by Rkczki [8]) nd Pintér nd Rkczki [6]), for k = 1 nd 3 we obtin effective sttements

3 ON EQUAL VALUES OF POWER SUMS OF ARITHMETIC PROGRESSIONS 55 Theorem 13 For k = 1 nd l / {1,3,5}, the eqution 16) S 1,b x) = Sl c,d y) implies mx x, y ) < C 1, where C 1 is n effectively computble constnt depending only on,b,c,d nd l In the exceptionl cses l = 3,5 one cn give some vlues for,b,c,d such tht the corresponding equtions possess infinitely mny solutions For exmple, if k = 1, =,b = 1,l = 3 or l = 5,c = 1,d = 0 we hve x = x 1 = y 1) 3 or x = x 1 = y 1) 5, respectively These equtions hve infinitely mny integer solutions, see [9] Theorem 14 For k = 3 nd l / {1,3,5}, the eqution 17) S 3,b x) = Sl c,d y) implies mx x, y ) < C, where C is n effectively computble constnt depending only on,b,c,d nd l Auxiliry results In this section, we collect some results needed to prove Theorem 1 First, we recll the finiteness criterion of Bilu nd Tichy []) To do this, we need to define five kinds of so-clled stndrd pirs of polynomils Let α,β be nonzero rtionl numbers, µ,ν,q > 0 nd ρ 0 be integers, nd let νx) Q[x] be nonzero polynomil which my be constnt) A stndrd pir of the first kind is x q,αx ρ νx) q ) or switched, αx ρ νx) q, x q ), where 0 ρ < q,gcdρ,q) = 1 nd ρ+degνx) > 0 A stndrd pir of the second kind is x,αx +β)νx) ) or switched DenotebyD µ x,δ)theµ-thdicksonpolynomil,definedbythefunctionl eqution D µ z +δ/z,δ) = z µ +δ/z) µ or by the explicit formul D µ x,δ) = µ/ i=0 d µ,i x µ i with d µ,i = µ µ i µ i i ) δ) i A stndrd pir of the third kind is D µ x,α ν ),D ν x,α µ )), where gcdµ,ν) = 1 A stndrd pir of the fourth kind is α µ/ D µ x,α), β ν/ D ν x,β)), where gcdµ,ν) = A stndrd pir of the fifth kind is αx 1) 3,3x 4 4x 3 ) or switched

4 56 A BAZSÓ, D KRESO, F LUCA AND Á PINTÉR The following theorem is the min result of [] Theorem 1 Let Rx), Sx) Q[x] be nonconstnt polynomils such tht the eqution Rx) = Sy) hs infinitely mny solutions in rtionl integers x,y Then R = ϕ f κ nd S = ϕ g λ, where κx),λx) Q[x] re liner polynomils, ϕx) Q[x], nd fx),gx)) is stndrd pir The following lemms re the min ingredients for the proofs of Theorems 13 nd 14 Lemm For every b Q nd rtionl integer k 3 with k / {4,6} the polynomil B k x)+b hs t lest three zeros of odd muliplicities Proof For b = 0 nd odd vlues of k 3 this result is consequence of theorem by Brillhrt [4, Corollry of Theorem 6]) For non-zero rtionl b nd odd k with k 3 nd for even vlues of k 8 our lemm follows from [6, Theorem] nd [8, Theorem 3], respectively Our next uxiliry result is n esy consequence of n effective theorem concerning the S-integer solutions of so-clled hyperelliptic equtions Lemm 3 Let fx) be polynomil with rtionl coefficients nd with t lest three zeros of odd multiplicities Further, let u be fixed positive integer If x nd y re integer solutions of the eqution x f = y u), then we hve mx x, y ) < C 3, where C 3 is n effectively computble constnt depending only on u nd the prmeters of f Proof This is specil cse of the min result of [5] Let c 1,e 1 Q nd c 0,e 0 Q Lemm 4 The polynomil S k,b c 1x+c 0 ) is not of the form e 1 x q +e 0 with q 3 Lemm 5 The polynomil S k,b c 1x+c 0 ) is not of the form e 1 D ν x,δ)+e 0, where D ν x,δ) is the ν-th Dickson polynomil with ν > 4,δ Q Before proving the bove lemms, we introduce the following nottion Put S k,b c 1x+c 0 ) = s k+1 x k+1 +s k x k + +s 0, nd c 0 = b +c 0

5 ON EQUAL VALUES OF POWER SUMS OF ARITHMETIC PROGRESSIONS 57 We hve 8) s k+1 = k c k+1 1 k +1, 9) 10) nd for k 4, s k = k c k 1 c 0 1), s k 1 = k c1 k 1 k6c 0 6c 0 +1),k, 1 11) s k 3 = k c1 k 3 kk 1)k )30c c c 0 1) 70 Proof of Lemm 4 Suppose tht S,b k c 1x+c 0 ) = e 1 x q +e 0, where we hve q = k+1 3 It follows tht s k 1 = 0, so 6c 0 6c 0 +1 = 0 Hence, c 0 / Q, which is contrdiction Proof of Lemm 5 Suppose tht S k,b c 1x + c 0 ) = e 1 D ν x,δ) + e 0 with ν > 4 Then 1) 13) 14) s k+1 = e 1, s k = 0, s k 1 = e 1 νδ, s k 3 = e 1ν 3)νδ 15) From 8), 1) nd 9), 13), respectively, it follows tht 16) e 1 = ν 1 c ν 1 nd c 0 ν = 1 In view of 10), substituting 16) together with k = ν 1 into 14), we obtin 17) ν 1 c1 ν ν 1) = ν 1 c ν 1 νδ, 4 ν which implies 18) c 1 = ν 1 4δ Similrly, compringtheforms11)nd15)ofs k 3 withthesubstitutions k = ν 1 nd 16), we obtin 7 ν 1 c1 ν 4 ν 1)ν )ν 3) 19) = ν 1 c ν 1ν 3)νδ, 5760 ν which implies 0) c 4 1 = 7ν 1)ν ) 880δ

6 58 A BAZSÓ, D KRESO, F LUCA AND Á PINTÉR After substituting 18) into 0), we obtin 7ν ) = 5ν 1), which implies ν = 9/, contrdiction nd or One cn see tht the condition ν > 4 is necessry Indeed, S,1 x) = 4 3 x3 1 3 x = 4 3 D 3 x, 1 ), 1 S,1 3 x) = x4 x = D 4 x, 1 ) Proofs of the Theorems Proof of Theorem 13 Using 3), one cn rewrite eqution 6) s c l B l+1 y + d ) )) d B l+1 = 1 l+1 c c x + b ) x 8c l B l+1 y + d ) )) d B l+1 = 4 x +8 b ) x l+1 c c = x+b ) b ) Then our result is simple consequence of Lemms nd 3 Proof of Theorem 13 Following Theorem 11, we hve S,bx) 3 = 3 x+ b 4 1 ) 4 3 x+ b 8 1 ) b +3b 3 16b 4 64 Using the bove representtion, we rewrite eqution 7) s 64S l c,d y) = x+b )4 4 x+b ) b +3b 3 16b 4 or 64S l c,d y) b 3b 3 16b 4 = X ), where X = x + b ) As in the previous cse, Lemms nd 3 complete the proof Proof of Theorem 1 If the eqution 5) hs infinitely mny integer solutions, then by Theorem 1 it follows tht S k,b 1x+ 0 ) = ϕfx)) nd S l c,d b 1x+b 0 ) = ϕgx)), where f,g) is stndrd pir over Q, 0, 1,b 0,b 1 rertionlswith 1 b 1 0 nd ϕx) is polynomil with rtionlcoefficients Assume tht h = degϕ > 1 Then Theorem 11 implies 0 < degf,degg,

7 ON EQUAL VALUES OF POWER SUMS OF ARITHMETIC PROGRESSIONS 59 nd since k < l, we hve degf = 1,degg = In prticulr, k + 1 = h nd l + 1 = h, so l = k + 1 Therefore, if l k + 1, we then must hve h = degϕ = 1 nd l = k +1 Condition k 1 implies k nd since l = k+1, it follows tht l 5 Sincedegf = 1,thereexistf 1,f 0 Q, f 1 0,suchthtS k,b f 1x+f 0 ) = ϕx), so S k,bf 1 gx)+f 0 ) = ϕgx)) = S l c,db 1 x+b 0 ) As gx) is qudrtic, by mking the substitution x x b 0 )/b 1, we obtin tht there re c,c 1,c 0 Q, c 0, such tht S k,b c x +c 1 x+c 0 ) = S l c,d x) Since degs k,b x) = k+1 nd c 0, we hve decomposition of S l c,d x) whichisequivlenttosx+b/ 1/) )forsomes Q[x] withdegs = k+1, ccording to Theorem 11 Therefore, there exists liner polynomil lx) in C[x] such tht c x +c 1 x+c 0 = lx+b/ 1/) ) nd Sx) = S,b k lx)) Hence, there re A,B C, A 0, such tht c x +c 1 x+c 0 = Ax+b/ 1/) +B Clerly, this implies tht A,B Q nd S,b k Ax+b/ 1/) +B ) = S k+1 c,d x) By the liner substitution x x b/+1/, we obtin 31) S k,b Ax +B) = S k+1 c,d x b/+1/) Thus, we hve n equlity of polynomils of degree k+ 6 We clculte nd compre coefficients of the first few highest monomils prticipting in the bove polynomils The coefficients of the polynomil in the right hnd side bove re esily deduced by setting c 1 = 1,c 0 = b/ + 1/ in 8), 9), 10) nd 11) Therefore, if we denote nd S k+1 c,d x b/+1/) = r k+ x k+ + +r 1 x+r 0, c 0 = d c b + 1,

8 60 A BAZSÓ, D KRESO, F LUCA AND Á PINTÉR then the coefficients re: r k+ = ck+1 k +, r k+1 = ck+1 c 0 1), r k = ck+1 k +1) 6c 0 1 6c 0 +1), r k = ck+1 k +1)kk 1) 30c c c 0 1) 70 On the other hnd, the coefficients s k+1,s k,s 0 for the polynomil S k,b x) cn be found by setting c 1 = 1,c 0 = 0 in 8), 9), 10) nd 11) Since S k,b Ax +B) = it follows tht if we put then k+1 m=0 s m m i=0 ) m Ax ) i B m i, i S k,b Ax +B) = t k+ x k+ + +t 1 x+t 0, t k+ = k A k+1 k +1, t k+1 = 0, t k = k A k B + k A k ) ) b 1, t k 1 = 0, t k = k k Ak 1 B + k k Ak 1 B + k k 1 Ak 1 6 b ) 6 b ) b ) ) +1 ) 1 Now we compre the coefficients Compring the leding coefficients yields 3) nd Therefore, k A k+1 k +1 = ck+1 k+, so k A k+1 = c k+1, c = ck+ k+1 A k+1 k+1 c Q

9 ON EQUAL VALUES OF POWER SUMS OF ARITHMETIC PROGRESSIONS 61 If nd c do not fulfill the bove condition, we re through, otherwise we proceed Compring the coefficients of index k +1, we get c k+1 c 0 1) = 0, so c 0 = 1/, which implies d c = b If the coefficients,b,c nd d do not stisfy the lst property bove, then we eliminte the possibility degϕ > 1 Therefore, we proceed with the cse where,b,c nd d do stisfy this property Compring the next coefficients nd using 3), we obtin b 33) 1 = 1 Ak +1) B 1 Compring the coefficients of index k nd using c 0 = 1/, we get k ) ) k Ak 1 B + k k b Ak 1 B 1 ) ) ) + k k b b 1 Ak = 7 8 ck+1 k +1)kk 1) 70 By using lso 3) nd simplifying, we obtin B + B ) ) b ) ) ) b b = 74k 1)A 1440 By using lso 33), the lst reltion bove cn be trnsformed into B +B 1 ) Ak +1) B ) 1 1 Ak +1) B 1 4 After simplifiction, we obtin = 7A 4k 1) 1440 A k 3) k 1) = 15 For k 3, the expression in the left hnd side bove is negtive or zero, which is contrdiction If k =, then A = 3, which contrdicts the fct tht A Q Therefore there re no rtionl coefficients,b,c,d,a nd B such tht 31) is fulfilled, which implies tht degϕ = 1

10 6 A BAZSÓ, D KRESO, F LUCA AND Á PINTÉR Now, we hve S k,b 1x+ 0 ) = e 1 fx)+e 0 nd S l c,d b 1x+b 0 ) = e 1 gx)+e 0, where 0 e 1,e 0 Q Further, we hve degf = k +1 nd degg = l+1 In view of the ssumptions on k nd l, it follows tht the stndrd pir f,g) cnnot be of the second kind, nd with the exception of the cse k,l) = 3,5), of the fifth kind either If it is of the first kind, then one of the polynomils S k,b 1x + 0 ) nd S l c,d b 1x + b 0 ) is of the form e 1 x q + e 0 with q 3 This is impossible by Lemm 4 If f,g) is stndrd pir of the third or fourth kind, we then hve S l c,d b 1x + b 0 ) = e 1 D ν x,δ) + e 0 with ν = l nd δ Q, which contrdicts Lemm 5 or k =,l = 3 In this cse Theorem 14 gives n effective finiteness result Now returning to the specil cse k,l) = 3,5), by using formul 10) for k = 3 it is esy to see tht S 3,b c 1c+c 0 ) = e 1 3x 4 4x 3 )+e 0 is impossible, see the proof of Lemm 4 Acknowledgements The uthors re grteful to the referee for her/his creful reding nd helpful remrks The reserch ws supported in prt by the Hungrin Acdemy of Sciences, by the OTKA grnts K75566, K100339, NK101680, NK10408 nd by the TÁMOP 41/B-09/1/KONV project implemented through the New Hungry Development Pln cofinnced by the Europen Socil Fund nd the Europen Regionl Development Fund Dijn Kreso ws supported by the Austrin Science Fund FWF): W130-N13 nd NAWI Grz References [1] A Bzsó, Á Pintér nd H M Srivstv, A refinement of Fulhber s Theorem concerning sums of powers of nturl numbers, Appl Mth Lett 5 01), [] Y F Bilu nd R F Tichy, The Diophntine eqution fx) = gy), Act Arith ), [3] Y F Bilu, B Brindz, P Kirschenhofer, Á Pintér nd R F Tichy, Diophntine equtions nd Bernoulli polynomils with n Appendix by A Schinzel), Compositio Mth ), [4] J Brillhrt, On the Euler nd Bernoulli polynomils, J Reine Angew Mth ), [5] B Brindz, On S-integrl solutions of the eqution y m = fx), Act Mth Hungr ), [6] Á Pintér nd Cs Rkczki, On the zeros of shifted Bernoulli polynomils, Appl Mth Comput ), [7] H Rdemcher, Topics in Anlityc Number Theory, Springer-Verlg, Berlin, 1973

11 ON EQUAL VALUES OF POWER SUMS OF ARITHMETIC PROGRESSIONS 63 [8] Cs Rkczki, On some generliztions of the Diophntine eqution s1 k + k + + x k )+r = dy n Act Arith ), [9] J J Schäffer, The eqution 1 p + p + 3 p + + n p = m q, Act Mth ), A Bzsó Institute of Mthemtics MTA-DE Reserch Group Equtions, functions nd curves Hungrin Acdemy of Science University of Debrecen H-4010 Debrecen, PO Box 1 Hungry E-mil: bzso@scienceunidebhu D Kreso Institut für Mthemtik A) Technische Universität Grz Steyrergsse 30, 8010 Grz Austri E-mil: kreso@mthtugrzt F Luc Mthemticl Center UNAM UNAM Ap Postl 61 3 Xngri) CP , Moreli, Michocán Mexico E-mil: fluc@mtmorunmmx Á Pintér Institute of Mthemtics MTA-DE Reserch Group Equtions, functions nd curves Hungrin Acdemy of Science University of Debrecen H-4010 Debrecen, PO Box 1 Hungry E-mil: pinter@scienceunidebhu Received: Revised: 71011