1. introduction Stancliff [16] noted without proof an interesting property of the Fibonacci. In 1980 Winans [23] investigated the related sums.

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1 ON THE LUCAS SEQUENCE EQUATION U n SZ TENGELY U k x k Abstract In 953 Stancliff noted an interesting property of the Fibonacci number F 89 One has that , where in the numerators the elements of the Fibonacci sequence appear We provide methods to determine similar identities in case of Lucas sequences As an example we prove that U k0 U k 647 k+, where U 0 0, U and U n 4U n + U n 2, n 2 introduction Stancliff [6] noted without proof an interesting property of the Fibonacci sequence F n One has that F F k 0 k+ k0 In 980 Winans [23] investigated the related sums k0 F αk 0 k+ for certain values of α In 98 Hudson and Winans [8] provided a complete characterization of all decimal fractions that can be approximated by sums of the type F α n F αk, α, l 0l(k+) 2000 Mathematics Subject Classification Primary D25; Secondary B39 Key words and phrases Lucas sequences, Diophantine equations, Elliptic curves

2 2 SZ TENGELY Long [2] proved a general identity for binary recurrence sequences from which one obtains eg 9899 k0 F k 0 2(k+), 09 k0 F k ( 0) k+ In the previous examples decimal fractions were studied, in case of different bases characterizations were obtained by Jia Sheng Lee [0] and by Köhler [9] and by Jin Zai Lee and Jia Sheng Lee [] Here we state a result by Köhler that we will use later in this article Theorem A Let A, B, a 0, a be arbitrary complex numbers Define the sequence {a n } by the recursion a n+ Aa n + Ba n Then the formula a k x a 0x Aa 0 + a k+ x 2 Ax B k0 holds for all complex x such that x is larger than the absolute values of the zeros of x 2 Ax B Let P and Q be non-zero relatively prime integers The Lucas sequence {U n (P, Q)} is defined by U 0 0, U and U n P U n QU n 2, if n 2 In this paper we deal with the determination of all integers x 2 for which there exists an n 0 such that U k () U n x, k where U n is a Lucas sequence with some given P and Q In case of P, Q one gets the Fibonacci sequence De Weger [5] computed all x 2 in case of the Fibonacci sequence, the solutions are as follows F F 2 F 0 55 F k 2 k, F 5 5 F k 8 k, F 89 F k 3 k, F k 0 k De Weger applied arguments of algebraic number theory and obtained two Thue equations, which were solved using Baker s method (see eg [2, 7, 5]) In the current work, we show how to reduce a search for U k x k integral x 2 related to the equation U n to elliptic Diophantine equations or to Thue equations following an elementary argument by Alekseyev and Tengely [] There exists a number of

3 ON THE LUCAS SEQUENCE EQUATION 3 software implementations for determining integral points on elliptic curves, these procedures are based on a method developed by Stroeker and Tzanakis [8] and independently by Gebel, Pethő and Zimmer [6] The elliptic logarithm method for determining all integer points on an elliptic curve has been applied to a variety of elliptic equations (see eg [9, 20, 2, 22]) 2 main results Theorem Let {U n (P, Q)} be a Lucas sequence with Q {±} and (P, Q) / {( 2, ), (2, )} Then equation () possesses only finitely many solutions in n, x which can be effectively determined The proof of Theorem provides an algorithm to determine all solutions of equation () Following this algorithm we obtain numerical results Theorem 2 Let {U n (P, Q)} be a Lucas sequence with 0 P 0, Q {±} and (P, Q) ( 2, ), (2, ) Then equation () has the following solutions (P, Q, n, x) {( 3,, 5, 6), (,, 5, 2), (,,, 9), (,,, 2), (,, 2, 2), (,, 5, 3), (,, 0, 8), (,,, 0), (2,, 2, 3), (3,, 2, 4), (3,,, 3), (3,, 5, 9), (4,, 2, 5), (4,, 0, 647), (4,,, 4), (5,, 2, 6), (5,,, 5), (6,, 2, 7), (6,,, 6), (7,, 2, 8), (7,,, 7), (8,, 2, 9), (8,,, 8), (9,, 2, 0), (9,,, 9), (0,, 2, ), (0,,, 0)} Remark We note that the sequences with (P, Q) {( 2, ), (2, )} are degenerate sequences, that is the discriminant D P 2 4Q is zero One has that U n ( 2, ) ( ) n+ n and U n (2, ) n In these cases there are infinitely many solutions of equation (): (x + ) 2 U (x+) 2( 2, ) (x ) 2 U (x ) 2(2, ) U k ( 2, ), if x is even, x k U k (2, ) x k 3 auxiliary results The Lucas sequences {U n (P, Q)} and associated Lucas sequences {V n (P, Q)} are defined by the same linear recurrent relation with P, Q

4 4 SZ TENGELY Z \ {0} but different initial terms: U 0 0, U and U n P U n QU n 2, if n 2, V 0 2, V P and V n P V n QV n 2, if n 2 Terms of Lucas sequences and associated Lucas sequences satisfy the following identity (2) V 2 n DU 2 n 4Q n, where D P 2 4Q To determine the appropriate Thue equations we use parametric solutions of ternary quadratic equations Such parametrization are given in [, Theorem ] Lemma Let A, B, C be non-zero integers and let (x 0, y 0, z 0 ) with z 0 0 be a particular non-trivial integer solution to the Diophantine equation Ax 2 + By 2 + Cz 2 0 Then its general integer solution is given by (x, y, z) p q (P x(m, n), P y (m, n), P z (m, n)) where m, n as well as p, q are coprime integers with q > 0 dividing 2 lcm(a, B)Cz 2 0, and P x (m, n) x 0 Am 2 + 2y 0 Bmn x 0 Bn 2, P y (m, n) y 0 Am 2 + 2x 0 Amn + y 0 Bn 2, P z (m, n) z 0 Am 2 + z 0 Bn 2 4 proofs Proof of Theorem Let {U n (P, Q)} be a Lucas sequence with Q {±} Theorem A implies that U k (3) x k x 2 P x ± Hence from equations () and (3) it follows that U n x 2 P x ± Finiteness follows from results by Nemes and Pethő [3] and by Pethő [4] We provide two approaches to determine a finite set of possible values of x for which U n x 2 P x± The first one is based on elliptic curves It only works if one can determine the ranks of the appropriate Mordell-Weil groups The second method is based on an elementary reduction algorithm which yield finitely many quartic Thue equations to solve After computing the finite set of possible values of x we use eigenvalues and eigenvectors to compute the sum U k x k

5 ON THE LUCAS SEQUENCE EQUATION 5 Substituting U n x 2 P x ± into the identity (2) yields a genus curve C (P,Q,n) : y 2 (P 2 4Q)(x 2 P x + Q) 2 + 4Q n Bruin and Stoll [4] described and algorithmized the so-called two-cover descent, which can be used to prove that a given hyperelliptic curve has no rational points This algorithm is implemented in Magma [3], the procedure is called TwoCoverDescent If it turns out that there are no rational points on the curves y 2 (P 2 4)(x 2 P x + ) and y 2 (P 2 + 4)(x 2 P x ) 2 ± 4, then equation () has no solution If TwoCoverDescent yields that rational points may exist, but the procedure Points fails to find one, then we follow the second approach, solution via Thue equations that we consider later in the proof Now we assume that we could determine points on curves for which TwoCoverDescent predicts existence of rational points That means we are given elliptic curves in quartic model Tzanakis [2] provided a method to determine all integral points on quartic models, the algorithm is implemented in Magma as IntegralQuarticPoints The second approach is based on Lemma There are three ternary quadratic equations to parametrize Q : X 2 (P 2 4)Y 2 4Z 2 0, Q 2 : X 2 (P 2 + 4)Y 2 4Z 2 0, Q 3 : X 2 (P 2 + 4)Y 2 + 4Z 2 0 There are points on these curves: Q : (X, Y, Z) (2, 0, ), Q 2 : (X, Y, Z) (2, 0, ), Q 3 : (X, Y, Z) (P,, ) It follows from Lemma that ± P z (m, n) p q (m2 (P 2 ± 4)n 2 ) Therefore p ± We deal with the curve y 2 (P 2 4)(x 2 P x + ) 2 + 4, the other two cases are similar We obtain that Hence we have that x 2 P x + ±4mn q q(2x P ) 2 ± (4 P 2 )(m 2 (P 2 4)n 2 ) 6mn 0, where q > 0 divides 8(P 2 4) Applying Lemma again we obtain that m f m (u, v) and n f n (u, v), where f m, f n are homogeneous quadratic polynomials It remains to compute integral solutions of

6 6 SZ TENGELY finitely many quartic Thue equations (it may happen that the defining quartic polynomials are reducible, in those cases the resolution is simpler) ±q f m (u, v) 2 (P 2 4)f n (u, v) 2 If we have a possible solution x N of equation (), then we have to compute the value of the sum U k We define x ( k ) P/x Q/x T /x 0 Following standard arguments one has that ( T 0 + T + T T N ) ( ) ( x 0 N U k x k Using eigenvectors and eigenvalues one can determine a formula for the powers of T, hence one obtains a formula depending only on N for the sum N U k To find solutions of the equation () it remains to x k compute U k x k ) Proof of Theorem 2 We will illustrate how one can use the approaches provided in the proof of Theorem to determine all solutions of equation () for given P and Q First we deal with the case P 4, Q We have that y 2 20(x 2 4x ) 2 ± 4 To determine all integral solutions we use the Magma commands IntegralQuarticPoints([20, -60, 280, 60, 6]) and IntegralQuarticPoints([20, -60, 280, 60, 24],[-,-8]) One obtains that x { 643,, 0,, 3, 4, 5, 647} Since x 2 only 4 values remain In case of x 647 the matrix T is as follows ( ) 4/647 /647 /647 0 and we obtain that ( 2 ) N ( 5 U k (29 5 ) 2+ ) N ( ) k

7 Thus ON THE LUCAS SEQUENCE EQUATION 7 In a similar way we get that U k 647 k U 0 U k 3 k +, U k 4 k +, U k 5 k 4 U 2 We apply the second method to completely solve equation () with P 3, Q The curve C P,Q,n has the form y 2 5(x 2 3x + ) It can be written as v 2 5u 4 50u with v 4y and u 2x 3 The second approach has been implemented in Sage [7] by Alekseyev and Tengely [] Using their procedure QuarticEq([5,-50,89]) we obtain that u {±, ±3, ±5}, therefore x { 6, 0,, 2, 3, 9} We have that U k 2 k +, U k 3 k U, U k 9 k 55 U 5 Acknowledgement The research was supported by the European Union and the State of Hungary, co-financed by the European Social Fund in the framework of TÁMOP 424 A/ National Excellence Program References [] M A Alekseyev and Sz Tengely On integral points on biquadratic curves and near-multiples of squares in lucas sequences J Integer Seq, 7(6):Article 466, 5, 204

8 8 SZ TENGELY [2] A Baker Contributions to the theory of Diophantine equations I On the representation of integers by binary forms Philos Trans Roy Soc London Ser A, 263:73 9, 967/968 [3] W Bosma, J Cannon, and C Playoust The Magma algebra system I The user language J Symbolic Comput, 24(3-4): , 997 Computational algebra and number theory (London, 993) [4] N Bruin and M Stoll Two-cover descent on hyperelliptic curves Math Comp, 78(268): , 2009 [5] B M M de Weger A curious property of the eleventh Fibonacci number Rocky Mountain J Math, 25(3): , 995 [6] J Gebel, A Pethő, and H G Zimmer Computing integral points on elliptic curves Acta Arith, 68(2):7 92, 994 [7] K Győry Solving Diophantine equations by Baker s theory In A panorama of number theory or the view from Baker s garden (Zürich, 999), pages Cambridge Univ Press, Cambridge, 2002 [8] R H Hudson and C F Winans A complete characterization of the decimal fractions that can be represented as 0 k(i+) F αi, where F αi is the αith Fibonacci number Fibonacci Quart, 9(5):44 42, 98 [9] G Köhler Generating functions of Fibonacci-like sequences and decimal expansions of some fractions Fibonacci Quart, 23():29 35, 985 [0] Jia Sheng Lee A complete characterization of B-power fractions that can be represented as series of general Fibonacci numbers or of general Tribonacci numbers Tamkang J Management Sci, 6():4 52, 985 [] Jin Zai Lee and Jia Sheng Lee A complete characterization of B-power fractions that can be represented as series of general n-bonacci numbers Fibonacci Quart, 25():72 75, 987 [2] C T Long The decimal expansion of /89 and related results Fibonacci Quart, 9():53 55, 98 [3] I Nemes and A Pethő Polynomial values in linear recurrences II J Number Theory, 24():47 53, 986 [4] A Pethő On the solution of the equation G n P (x) In Fibonacci numbers and their applications (Patras, 984), volume 28 of Math Appl, pages Reidel, Dordrecht, 986 [5] N P Smart The algorithmic resolution of Diophantine equations, volume 4 of London Mathematical Society Student Texts Cambridge University Press, Cambridge, 998 [6] F Stancliff A curious property of a ii Scripta Math, 9:26, 953 [7] W A Stein et al Sage Mathematics Software (Version 62) The Sage Development Team, [8] R J Stroeker and N Tzanakis Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms Acta Arith, 67(2):77 96, 994 [9] R J Stroeker and N Tzanakis Computing all integer solutions of a general elliptic equation In Algorithmic number theory (Leiden, 2000), volume 838 of Lecture Notes in Comput Sci, pages Springer, Berlin, 2000 [20] R J Stroeker and N Tzanakis Computing all integer solutions of a genus equation Math Comp, 72(244):97 933, 2003

9 ON THE LUCAS SEQUENCE EQUATION 9 [2] N Tzanakis Solving elliptic Diophantine equations by estimating linear forms in elliptic logarithms The case of quartic equations Acta Arith, 75(2):65 90, 996 [22] N Tzanakis Effective solution of two simultaneous Pell equations by the elliptic logarithm method Acta Arith, 03(2):9 35, 2002 [23] C F Winans The Fibonacci series in the decimal equivalents of fractions In A collection of manuscripts related to the Fibonacci sequence, pages 78 8 Fibonacci Assoc, Santa Clara, Calif, 980 Mathematical Institute University of Derecen POBox Debrecen Hungary address: tengely@scienceunidebhu