Egyetemi doktori (PhD értekezés tézisei Combinatoria Diophantine equations Kombinatorikus diofantikus egyenetek Kovács Tünde Témavezet : Dr. Hajdu Lajos Debreceni Egyetem Matematika- és Számítástudományok Doktori Iskoa Debrecen, 2011.
Egyetemi doktori (PhD értekezés tézisei Combinatoria Diophantine equations Kombinatorikus diofantikus egyenetek Kovács Tünde Témavezet : Dr. Hajdu Lajos Debreceni Egyetem Matematika- és Számítástudományok Doktori Iskoa Debrecen, 2011.
Introduction Our dissertation consists of ve chapters each containing new resuts concerning Diophantine equations and Diophantine probems. Most probems have certain combinatoria background. These resuts have been pubished in our papers [20], [25], [26], [27] and [21], respectivey. Here we sha give an overview of the contents of the chapters, but before doing so we make some introductory notes on the subjects of our thesis. In the rst chapter we present the main method used in our studies namey the Eog method together with an improvement due to Hajdu and Kovács [20]. For the resoution of eiptic equations, an ecient method was deveoped simutaneousy and independenty by Stroeker, Tzanakis [47] and Gebe, Peth, Zimmer [14] in 1994. The most recent version of this so-caed Eog method is aready capabe to nd (at east in principe a integra points on genus 1 curves (see [49], and aso the references given there. Furthermore, Stroeker and Tzanakis [48] provided an agorithm to determine that Morde-Wei basis of the curve in which one can get the best bound for the integra points of the curve. In the rst chapter we show that working with more Morde-Wei bases simutaneousy, the na bound for the integra points can be further decreased. The resuts of this chapter are pubished in [20]. In the second chapter we appy the Eog method to sove many combinatoria Diophantine equations whose soutions are missing from the iterature. One of the rst resuts giving a integer soutions of a combinatoria Diophantine equation is a theorem of Morde [35], which provides a integer soutions of the equation y(y + 1 = ( + 1( + 2. Other scattered equations have been investigated by severa authors, see for eampe [1], [2], [9], [31], [43], [47], [57], [58], [61]. Hajdu and Pintér [22] systematicay coected and soved those combinatoria equations that can be reduced to Morde-type equations. We etend this resut to more genera combinatoria equations that can be reduced to genera eiptic equations. Namey, we coect those equations that can be reduced to equations of genus 1. The resuts of the second chapter are pubished in [25]. In the third chapter we give severa eective and epicit resuts concerning the vaues of some poynomias in binary recurrence sequences. First we provide an eective niteness theorem for certain combinatoria numbers (binomia coef- cients, products of consecutive integers, power sums, aternating power sums in binary recurrence sequences, under some assumptions. The proof of this theorem is based on Baker's method. We aso give an ecient agorithm (based on genus 1 curves for determining the vaues of certain degree 4 poynomias in 1
such sequences. Finay, party by the hep of this agorithm we competey determine a combinatoria numbers of the above type for the sma vaues of the parameter invoved in the Fibonacci, Lucas, Pe and Associated Pe sequences. The resuts of the third chapter are pubished in [26]. Baancing numbers and their generaizations have been investigated by severa authors, from many aspects. In the fourth chapter we introduce the concept of baancing numbers in arithmetic progressions, and prove severa eective niteness and epicit resuts about them. In the proofs of our resuts, among others, we combine Baker's method, the moduar method, the Chabauty method and the theory of eiptic curves. The resuts of the fourth chapter are pubished in [27]. In the fth chapter we prove that the product of k consecutive terms of a primitive arithmetic progression is never a perfect fth power when 3 k 54. This probem is cosey reated to a cassica probem, considered by severa mathematicians e.g. Erd s, Sefridge, Gy ry, Saradha, Shorey, Tijdeman. We aso provide a more precise statement, concerning the case where the product is an "amost" fth power. Our theorems yied considerabe improvements and etensions, in the fth power case, of recent resuts due to Gy ry, Hajdu and Pintér [19]. Whie the earier resuts have been proved by cassica (mainy agebraic number theoretica methods, our proofs are based upon a new too: we appy genus 2 curves and the Chabauty method (both the cassica and the eiptic verison. The resuts of the fth chapter are pubished in [21]. The Eog method and an improvement Eiptic Diophantine equations have a ong history. Even the eective theory of such equations have an etremey rich iterature. A cassica resut of Baker [3] yieds that an eiptic equation can have ony nitey many integer soutions, and the size (absoute vaue of the soutions can be eectivey bounded. However, to epicity nd a integra soutions another method has been deveoped, which uses the arithmetic properties of eiptic curves. This agorithm combines many deep ingredients, due to severa authors. Here we ony refer to the papers of Stroeker, Tzanakis [47] and Gebe, Peth, Zimmer [14], where the rst compete versions of this method are independenty given. (See aso the references in these papers. The most recent version of this so-caed Eog method is aready capabe to nd (at east in principe a integra points on genus one curves (see [49], and aso the references given there for certain important intermediate steps. To summarize the method in one sentence, what happens is that rst 2
the maimum N of the coecients of the integra points (in some Morde-Wei basis is bounded, and then this bound is graduay decreased to a size where the actua points can aready be identied by an ehaustive search. (Of course, in fact the method is much more genera and compicated. To get the na bound N fina for N, the LLL-agorithm is appied. First we briey summarize the main steps of the Eog method. We foow the presentation in [49]. Let f Z[u, v] and dene the curve C by C : f(u, v = 0. Suppose that C is of genus one. Then C is birationay equivaent (over a number ed to some eiptic curve E : 3 + A + B = y 2 with A, B Q. Let r be the rank of E, and et P 1,..., P r be a Morde-Wei basis of E. Then any rationa point P of E can be written as P = P 0 + n 1 P 1 +... + n r P r, (1 where P 0 is some torsion point of E, and n i Z (i = 1,..., r. Now on the one hand, using estimates of David [11] concerning inear forms in eiptic ogarithms, one gets a ower bound of the form L(P ep( c 1 (og N + c 2 (og og N + c 3 r+3. (2 Here L(P is a certain inear form in eiptic ogarithms (roughy" the eiptic ogarithm of P, N = ma n i and c 1, c 2, c 3 are constants depending ony on 1 i r the curves C and E. On the other hand, supposing that P is the image of an integra point of C under the above birationa transformation, using standard arguments (incuding height estimates of points of E we get an inequaity of the form L(P c 4 ep( c 5 λn 2. (3 Here c 4, c 5 are again constants, which depend ony on C and E. Further, most importanty from our point of view, λ is the east eigenvaue of the height pairing matri of the basis P 1,..., P r occurring in (1. That is, λ certainy depends on the choice of the Morde-Wei basis. As it is demonstrated by Stroeker and Tzanakis [48], the size of λ has a great impact on the na bound N fina for N. We sha return here a itte ater. 3
Combining estimates (2 and (3 we get an initia upper bound N 0 for N. However, this upper bound is usuay etremey huge. Due to an observation of Stroeker and Tzanakis [48], N 0 shoud be around 10 (5r2 +5r+28/2. Hence to epicity determine the integra points on C, this initia bound N 0 shoud be reduced. This can be done by attice reduction techniques due to de Weger [59], based on the LLL-agorithm. We use a variant due to Tzanakis [56]. To appy this resut, one starts with (3, together with the inequaity N < N 0. Using the appropriate Proposition from Section 5 of Tzanakis [56], one gets a new ower bound of the shape N < c 6 c5 λ for N, where c 6 is an epicity computabe constant depending on some parameters of E, and aso on the ength of the shortest vector of an LLL-reduced basis of a certain attice. As one can see, this new bound is inear in λ 1/2, which shows the importance of this parameter. Stroeker and Tzanakis [48] have considered severa eampes which indicate this phenomenon in a rather convincing way. Summarizing the resuts in [48], to get the best possibe reduced bound N fina for N one shoud denitey choose an ST-basis (i.e. a basis for which the corresponding λ vaue is the greatest possibe of the curve E in (1. Subsequenty, Stroeker and Tzanakis [48] have aso worked out an ecient agorithm for nding an ST-basis of the curve. However, in the seque it turns out that the bound obtained by using an STbasis, can sti be improved further, if one works with severa Morde-Wei bases simutaneousy. It is important to note that foowing our method the use of more bases sha increase ony by a fraction the tota time needed to get a better N fina. Aready a sma gain in N fina may ead to a arge improvement in the searching time for nding the sma soutions - in particuar, if r is arge. The reason is simpy that the region where we have to ook for the sma soutions is of size (2N fina + 1 r. Note that a simiar size" notion was used aso in [48] to compare the na bounds obtained in dierent Morde-Wei bases. Now we briey outine how to work in severa bases simutaneousy. To epain our ideas in fact it is sucient to use two bases. So assume that B 1 = (P 1,..., P r is a Morde-Wei basis of E, and et S be an integra unimoduar matri of size r r. Let B 2 = (Q 1,..., Q r be the basis of E obtained from B 1 by using S as a basis transformation matri. Let P be a rationa point on E with the representation (1, and assume that we aso have P = Q 0 + m 1 Q 1 +... + m r Q r, (4 4
with some torsion point Q 0 and integers m 1,..., m r. Put M = ma m i, and 1 i r reca that by eementary inear agebra we have S 1 n 1. n r = m 1. m r. (5 This impies M sn, where s = S 1 is the row norm of S 1. (The row norm of a k type rea matri A = (a ij 1 i k is dened by A = ma a ij. 1 j 1 i k j=1 In particuar, this means that one does not have to go through the Eog method both for B 1 and B 2, it is sucient to use it with B 1 say. Indeed, take for eampe B 1 to be an ST-basis of E, and suppose that after appying the Eog method (together with the reduction stage we have the bound N < N fina. Then by M sn, we automaticay have M M 0 := sn fina. As s is typicay sma" (it wi be at most around ten, M 0 is not too arge - and of course, it can aso be reduced. Importanty, we can get the na bound M fina very easiy and quicky. The reason is that the reduction steps are dicut and time consuming ony if the initia bound is arge, as then e.g. high precision is needed. However, as s wi be sma, the reduction steps eading from M 0 to M fina are made very easiy. The na bounds N fina and M fina yied simutaneous upper bounds for the coecients of P, in two dierent bases. Combining these two bounds by (5, we can decrease the domain where the na search has to be done. As one may predict (which wi turn out to be true, the gain starts getting more and more signicant as the rank r is getting arger and arger. Strategy 1. We try to decrease N (1 (corresponding to fina B 1, componentwise. For this purpose, choose distinct indices i, j with 1 i, j r and a positive integer t, and consider the bases B 2 and B 3 obtained by repacing P i by P i +tp j and P i tp j in B 1, respectivey (eaving the other basis eements untouched. With the bases B 2 and B 3 the reduction process starts from the quite sma bound (t + 1N (1 fina N (3 fina (2 and gives, respectivey, the na bounds, say, N. Then a simpe cacuation yieds that n i N (2 fina + N (3 fina 2t fina and hods. If the right hand side happens to be ess than N (1 fina, then we get a new, improved bound for n i. To make this principe work, for each ed i we 5
(heuristicay choose that j, for which the sum of the λ vaues (corresponding to B 2 and B 3 is maima with t = 1. Then for simpicity (and aso because we try to keep the time consumption of the method ow, instead of checking severa vaues, we take the ed vaue t = 10 in the computations. The procedure can be iterated, and the iteration eads to further improvement in some cases. Further, we have some reasons for the choice of t = 10. If λ t denotes the λt λ t+1 vaue of λ corresponding to t (either in B 2 or in B 3, then t+1 t is cose to 1, if t is arge". The vaue t = 10 seems to be arge enough to make the λ-s corresponding to B 2 and B 3 more or ess cose to each other, and it seems to have some good eect on the outcome. Sti, obviousy at this point the method can have many variants. Strategy 2. Using the agorithm of Stroeker and Tzanakis [48], we determine the best" ten Morde-Wei bases B j (j = 1,..., 10 i.e. ten Morde-Wei basis corresponding to the ten argest λ-vaues. (Note that by the agorithm we get a the basis transformation matrices with respect to B 1, as we, and aso that the cacuation of ten basis takes ony a itte etra time than cacuating ony B 1. Then we compute the initia upper bounds N (j 0 (j = 1,..., 10 for the coordinates of the integra points in these bases, respectivey. (As we mentioned, out of these ony the cacuation of N (1 0 is time consuming (but it has to be cacuated even if we use ony B 1, the other bounds come very quicky and easiy. Having these bounds, using the basis transformation matrices, we get severa etra information for the coecients of P in B 1. In fact we get a system of inequaities dening a conve body, which contains much ess integra points than the one impied by n i N (1 fina (i = 1,..., r. Finay, we mention that atogether it seems that Strategy 2 yieds more improvement than Strategy 1. In the dissertation we give severa eampes to iustrate how our method works. The resuts of the rst chapter are pubished in [20]. Combinatoria Diophantine equations of genus 1 Many Diophantine equations possess combinatoria background. One of the rst resuts giving a integer soutions of a combinatoria Diophantine equation is a theorem of Morde [35], which provides a integer soutions of the equation y(y + 1 = ( + 1( + 2. Other scattered equations have been investigated by severa authors, see for eampe [1], [2], [9], [31], [43], [47], [57], [58], [61]. 6
Hajdu and Pintér [22] systematicay coected and soved those combinatoria equations that can be reduced to Morde-type equations. Our purpose is to etend this resut to more genera combinatoria equations that can be reduced to genera eiptic equations. Namey, we coect those equations that can be reduced to equations of genus 1. The equations are soved with the Eog method and with the Magma [8] computationa agebra system. We mention that beside a ot of sparse resuts (see e.g. [40], [41], [43], [50] and [60], Stroeker and de Weger [51] soved a such equations invoving binomia coecients. We need some notation to formuate our resuts. For a n, N et S n ( = 1 n + 2 n +... + ( 1 n, Π n ( = ( + 1 ( + n 1. The formery soved Diophantine equations which can be reduced to eiptic equations concerning Π n (, S n ( and ( n, are the foowings: Π 2 (k = Π 3 ( (Morde [35], ( 2 = 3 (Avanesov [1], Π 2 (k = Π 6 ( (MacLeod and Barrodae [31], S 2 (k = ( 2 (Avanesov [2] and Uchiyama [58], Π 3 (k = Π 4 (, S 2 (k = ( 4 (Boyd and Kisievsky [9], 2 = Π3 ( (Tzanakis and de Weger [57], 2 = Π4 ( (Pintér [40], see aso [15], p. 225., ( 4 = 2 (Pintér [41] and de Weger [60], ( 3 = 4 (de Weger [61], 4 = Π2 (, Π 3 ( (Pintér and de Weger [43], m = Πn (, where (m, n = (3, 6; 3, 6 (Stroeker and de Weger [50], ( m = n, where (m, n = (2; 3, 4, 6, 8, (3; 4, 6, (4; 6, 8 (Stroeker and de Weger [51], S 5 (k = ( 2, S5 (k = ( 4, Sm (k = Π n (, where (m, n = (2, 5; 2, 4, m = Π n (, where (m, n = (2, 4; 6, (3, 6; 2, 4, Π 4 (k = Π 6 ( (Hajdu and Pintér [22]. Here and ater on (k, = (a 1,..., a n ; b 1,..., b m means that (k, can be any of the pairs (a i, b j, i {1,..., n}, j {1,..., m}. 7
Equation Soutions S 3 (k = Π 2 ( (k, = ( 1, 0; 1, 0 S 3 (k = Π 4 ( (k, = ( 1, 0; 3, 2, 1, 0 S 3 (k = Π 8 ( (k, = ( 1, 0; 7, 6, 5, 4, 3, 2, 1, 0 S 5 (k = ( 3 (k, = ( 1, 0; 0, 1, 2, ( 2, 1; 3 S 7 (k = ( 2 (k, = ( 1, 0; 0, 1, ( 2, 1; 1, 2 P 2 (k = Π 4 ( (k, = ( 1, 0; 3, 2, 1, 0 P 2 (k = Π 8 ( (k, = ( 1, 0; 7, 6, 5, 4, 3, 2, 1, 0 P 3 (k = Π 6 ( (k, = ( 2, 1, 0; 5, 4, 3, 2, 1, 0, (8; 6, 1 P 4 (k = Π 8 ( (k, = ( 3, 2, 1, 0; 7, 6, 5, 4, 3, 2, 1, 0 Tabe 1: Equations which can be soved by Runge's method Equation Soutions S 3 (k = ( 3 (k, = ( 1, 0; 0, 1, 2, ( 2, 1; 3 S 3 (k = ( 6 (k, = ( 1, 0; 0, 1, 2, 3, 4, 5, ( 2, 1; 1, 6 S 3 (k = Π 3 ( (k, = ( 1, 0; 2, 1, 0 S 3 (k = Π 6 ( (k, = ( 1, 0; 5, 4, 3, 2, 2, 1, 0 S 5 (k = ( 2 (k, = ( 1, 0; 0, 1, ( 2, 1; 1, 2, ( 4, 3; 23, 24, ( 9, 8; 351, 352 S 5 (k = ( 4 (k, = ( 1, 0; 0, 1, 2, 3, ( 2, 1; 1, 4 S 5 (k = Π 2 ( (k, = ( 1, 0; 1, 0 S 5 (k = Π 4 ( (k, = ( 1, 0; 3, 2, 1, 0 Tabe 2: Equations which can be reduced to Morde-type equations We mention that S n ( is a poynomia of degree n + 1, and Π n ( is a poynomia of degree n. For the sake of competeness we give a integer soutions of the investigated poynomia equations (athough the negative soutions do not have combinatoria meanings. Our resuts are summarized in the net theorem. We distribute the equations considered into three tabes, according to the methods used in their soutions. Theorem 1 (Kovács, 2008. A integra soutions of the equations in the rst coumns of Tabes 1-3 are eacty the ones appearing in the second coumns of the tabes, respectivey. The resuts of the second chapter are pubished in [25]. 8
Equation Soutions S 1 (k = ( 4 (k, = ( 21, 20; 7, 10, ( 6, 5; 3, 6, ( 2, 1; 1, 4, ( 1, 0; 0, 1, 2, 3 S 1 (k = ( 8 (k, = ( 1, 0; 0, 1, 2, 3, 4, 5, 6, 7, ( 2, 1; 1, 8, ( 10, 9; 3, 10;, ( 78, 77; 7, 14, ( 221, 220; 10, 17 S 1 (k = Π 4 ( (k, = ( 16, 15; 5, 2, ( 1, 0; 3, 2, 1, 0 S 1 (k = Π 8 ( (k, = ( 1, 0; 7, 6, 5, 4, 3, 2, 1, 0 S 2 (k = ( 3 (k, = ( 1, 0; 0, 1, 2, ( 2, 1, (1, 3 S 2 (k = ( 6 (k, = ( 1, 0; 0, 1, 2, 3, 4, 5, (1; 1, 6 S 2 (k = Π 3 ( (k, = ( 1, 0; 2, 1, 0 S 2 (k = Π 6 ( (k, = ( 1, 0; 5, 4, 3, 2, 1, 0 S 3 (k = ( 2 (k, = ( 4, 3; 8, 9, ( 2, 1; 1, 2, ( 1, 0; 0, 1 S 3 (k = ( 4 (k, = ( 2, 1; 1, 4, ( 1, 0; 0, 1, 2, 3 S 3 (k = ( 8 (k, = ( 1, 0; 0, 1, 2, 3, 4, 5, 6, 7, ( 3, 2; 2, 9, ( 2, 1; 1, 8 S 5 (k = ( 6 (k, = ( 1, 0; 0, 1, 2, 3, 4, 5, ( 2, 1; 1, 6 S 5 (k = Π 3 ( (k, = ( 1, 0; 2, 1, 0 S 5 (k = Π 6 ( (k, = ( 1, 0; 5, 4, 3, 2, 1, 0 S 7 (k = ( 4 (k, = ( 2, 1; 1, 4, ( 1, 0; 0, 1, 2, 3 S 7 (k = Π 2 ( (k, = ( 1, 0; 1, 0 S 7 (k = Π 4 ( (k, = ( 1, 0; 3, 2, 1, 0 2 = Π8 ( (k, = (0, 1; 7, 6, 5, 4, 3, 2, 1, 0 4 = Π4 ( (k, = (0, 1, 2, 3; 3, 2, 1, 0 4 = Π8 ( (k, = (0, 1, 2, 3; 7, 6, 5, 4, 3, 2, 1, 0 8 = Π2 ( (k, = (0, 1, 2, 3, 4, 5, 6, 7; 1, 0 = Π4 ( (k, = (0, 1, 2, 3, 4, 5, 6, 7; 3, 2, 1, 0 8 Tabe 3: Equations which can be reduced to genus 1 equations 9
Combinatoria numbers in binary recurrences Let U = {U n } n=0 be a binary recurrence sequence dened by the initia terms U 0, U 1 Z and the recurrence reation U n = AU n 1 + BU n 2 (n 2 where A, B are non-zero integers. Let α and β denote the zeros of the companion poynomia 2 A B of U. Further, et D = A 2 + 4B be the discriminant of U and a u = U 1 βu 0, b u = U 1 αu 0, C = a u b u = U 2 1 AU 0 U 1 BU 2 0. The sequence U is caed non-degenerate if C 0 and α/β is not a root of unity. It is we-known that if U is non-degenerate then for a n = 0, 1,... we have U n = a uα n b u β n. α β From this point on we assume that B = ±1 and that U is non-degenerate. Then as it is aso we-known, U has a so-caed associate sequence V = {V n } n=0 for which Vn 2 DUn 2 = 4C( B n (6 hods for a n = 0, 1,.... Observe that by our assumption B = ±1, we have ( B n = ±1. Further, note that V 0 = 2U 1 AU 0, V 1 = AU 1 + 2BU 0 and V satises the same recurrence reation as U. Beside deaing with genera sequences U we consider combinatoria numbers in certain specia famous sequences, too. Let F, L, P and Q denote the Fibonacci, Lucas, Pe and Associated Pe sequence, respectivey. These sequences are dened by F 0 = 0, F 1 = 1, F n = F n 1 + F n 2 (n 2, L 0 = 2, L 1 = 1, L n = L n 1 + L n 2 (n 2, P 0 = 0, P 1 = 1, P n = 2P n 1 + P n 2 (n 2, Q 0 = 1, Q 1 = 1, Q n = 2Q n 1 + Q n 2 (n 2. Now we give what kind of combinatoria numbers we are interested in. Beside binomia coecients, we consider power sums, aternating power sums and products of consecutive integers as we. In the previous chapter we dened these poynomias ecept for the aternating power sum. That poynomia is dened for a k, N as R k ( = 1 k + 2 k... + ( 1 1 ( 1 k. 10
We mention that R k ( is a poynomia of degree k. We use the previous notation. Further, reca that B = ±1 and U = {U n } n=0 is non-degenerate. A our resuts concern the equation U n = p( (7 in integers n, with n 0. For the sake of competeness we aso take care of the soutions with 0, athough these soutions usuay do not have combinatoria meanings. First we give an eective resut for the soutions of (7 which is vaid for genera U. Theorem 2 (Kovács, 2009. Let k 2 and p( be one of the poynomias S k 1 (, R k (, Π k (, ( k. If either k = 2 or p( is one of S2 (, Π 3 (, ( 3, then further assume that B = 1. Then the soutions n, of equation (7 satisfy ma (n, < c 0 (U, k, where c 0 (U, k is an eectivey computabe constant depending ony on U and k. Obviousy, the assumption k 2 cannot be omitted. The net proposition shows that the condition B = 1 in the specia cases of the theorem is necessary as we. Proposition 1. Let U be the sequence dened by B = 1 and by the vaues U 0, U 1, A given in the ith row of Tabe 4, for any i {1, 2, 3, 4, 5}. Further, et p( be a poynomia from the ast coumn of the ith row of Tabe 4. Then equation (7 has innitey many soutions. U 0 U 1 A p( 1 253 254 S 1 (, R 2 (, ( 2 2 506 254 Π 2 ( 1 3759787041401 3760028828350 S 2 ( 7770 455962704852690 58682458798 ( 3 46620 2735776229116140 58682458798 Π 3 ( Tabe 4: Settings where equation (7 has innitey many soutions 11
Remark. If (7 has innitey many soutions then these soutions have some specia structure. This structure has been described by Nemes and Peth [36], see Theorem 3 (cf. aso [37], [38]. Szaay [52] gave an agorithm for the resoution of (7 in the case when p( is a poynomia of degree 3. We etend this resut to the degree 4 case. For this purpose we need some further notation. Let p( Q[] be a poynomia of degree 4 and write p( = A 0 4 + A 1 3 + A 2 2 + A 3 + A 4. Suppose that the coecients of p fu the reations A 0 = a e, A 1 = 4ab e, A 2 = 6ab2 + c, A 3 = 4ab3 + 2bc, A 4 = ab4 + b 2 c + d, e e e with some integers a, b, c, d, e, ae 0. Then we have p( = a( + b4 + c( + b 2 + d. e Write 1 = + b and et y = V n where V = {V n } n=0 is the associate sequence of U. Then by (6 we get ( a y 2 4 D 1 + c 2 2 1 + d = 4C( B n, e which yieds where Y 2 = h 4 X 4 + h 3 X 3 + h 2 X 2 + h 1 X + h 0, (8 Y = ey, X = 2 1, h 4 = a 2 D, h 3 = 2acD, h 2 = (c 2 + 2adD, h 1 = 2cdD, h 0 = d 2 D + 4e 2 C( B n. Equation (18 in genera is of genus 1, therefore the Eog method can be used to determine a its integra soutions. In particuar, using the program package Magma, equation (18 can be soved competey in concrete cases. If h 0 is a perfect square then (18 can be soved by the procedure IntegraQuarticPoints. Putting together some toos and resuts about genus 1 curves, we give an ecient method for the resoution of (18 in the genera case. For the description of the method see the proof of Theorem 3. (Further, we impemented our agorithm in Magma, too. From the soutions, the vaues and the indices n can be easiy determined. Summarizing the above argument, we get 12
= F n L n P n Q n S 1( [33] [34] [32] (0, 1, (0, 2, (1, 1, (1, 2, (2, 2, (2, 3 S 2( [52] [52] [52] (0, 2, (1, 2 S 3( [52] [52] [52] T 2( T 4( Π 2( Π 3( Π 4( ( 2 ( 3 ( 4 (0, 0, (0, 1, (1, 1, (2, 1, (4, 3, (8, 7, (10, 11 (0, 0, (0, 1, (1, 1, (2, 1 (0, 1, (0, 0, (3, 2, (3, 1 (0, 2, (0, 1, (0, 0 (0, 3, (0, 2, (0, 1, (0, 0 (1, 1, (2, 3, (18, 107 (1, 1 [33] [34] [32] (0, 1, (0, 2, (1, 1, (1, 2 (1, 1 (0, 1, (1, 1, (2, 3 (0, 0, (0, 1, (1, 1 (0, 1, (0, 0, (2, 2, (2, 1, (4, 4, (4, 3 (0, 2, (0, 1, (0, 0 (0, 3, (0, 2, (0, 1, (0, 0 [53] [53] [53] [52] [52] (0, 0, (0, 1, (0, 2, (0, 3, (1, 1, (1, 4, (3, 2, (3, 5, (6, 5, (6, 8 (0, 1, (1, 1 (0, 1, (0, 2, (1, 1, (1, 2, (2, 2, (2, 3 (1, 1, (1, 4 Tabe 5: Soutions of equation (7 with the particuar settings Theorem 3 (Kovács, 2009. Using the previous notation, suppose that 8aDd(2ad c 2 64a 2 C ± e 2 c 4 D. Then equation (7 has ony nitey many soutions n, and these soutions can be eectivey determined. Our na resut about this probem competey describes the above type combinatoria numbers for the sma vaues of the parameter k in some weknown binary recurrence sequences. Theorem 4 (Kovács, 2009. Let U {F, L, P, Q}, p( {S 1 (, S 2 (, S 3 (, R 2 (, R 4 (, Π 2 (, Π 3 (, Π 4 (, ( ( 2, ( 3, 4 }. Then the soutions n, of equation (7 are eacty those contained in Tabe 5. The sign shows that the actua equation has no soution. Further, the references given in the tabe indicate that the corresponding equation was soved in the appropriate paper. Remark. The compete soution of the equation U n = R 3 ( remains open. 13
These equations are of genus 2 thus neither Szaay's method nor our agorithm given in the proof of Theorem 3 can be used to sove them. The resuts of the third chapter are pubished in [26]. Resuts on (a, b-baancing numbers A positive integer n is caed a baancing number if 1 + + (n 1 = (n + 1 + + (n + r hods for some positive integer r (see [4] and [13]. The sequence of baancing numbers is denoted by B m (m = 1, 2,.... As one can easiy check, we have B 1 = 6 and B 2 = 35. Note that by a resut of Behera and Panda [4], we have B m+1 = 6B m B m 1 (m > 1. In particuar, there are innitey many baancing numbers. The iterature of baancing numbers is very rich. In [28] and [29] Liptai proved that there are no Fibonacci and Lucas baancing numbers, respectivey. Later, Szaay [54] derived the same resuts by a dierent method. In [30] Liptai, Luca, Pintér and Szaay generaized the concept of baancing numbers in the foowing way. Let y, k, be ed positive integers with y 4. A positive integer with y 2 is caed a (k, -power numerica center for y if 1 k + + ( 1 k = ( + 1 + + (y 1. In [30] severa eective and ineective niteness resuts were proved for (k, - power numerica centers. Recenty, the "baancing" property has been investigated in recurrence sequences (see [7]. In this chapter we etend the concept of baancing numbers to arithmetic progressions. Let a > 0 and b 0 be coprime integers. If for some positive integers n and r we have (a + b + + (a(n 1 + b = (a(n + 1 + b + + (a(n + r + b then we say that an + b is an (a, b-baancing number. The sequence of (a, b- baancing numbers is denoted by B m (a,b (m = 1, 2,.... We mention that since = B m for a m, we obtain a generaization of baancing numbers. B (1,0 m 14
We prove severa eective niteness and epicit resuts concerning poynomia vaues in the sequences B m (a,b. That is, we consider the equation B (a,b m = f( (9 in integers m and with m 1, where f is some poynomia with rationa coecients, taking ony integra vaues at integers. To prove our theorems, beside the above mentioned resuts of Ping-Zhi [39], Pintér and Rakaczki [42] and Rakaczki [44], we further need the moduar method deveoped by Wies [62] and others and a deep resut of Bennett [5] concerning binomia Thue equations. From this point on, when we refer to equation (9 we aways assume that a and b are arbitrary, but ed coprime integers such that a > 0 and b 0. Our rst resut is the foowing. Theorem 5 (Kovács, Liptai, Oajos, 2010. Let f( be a monic poynomia with integer coecients, of degree 2. If a is odd, then for the soutions of (9 we have ma(m, < c 0 (f, a, b, where c 0 (f, a, b is an eectivey computabe constant depending ony on a, b and f. Our net resut concerns the case where f( = with some 2. In this case soving equation (9 is equivaent to nding (a, b-baancing numbers which are perfect powers. Theorem 6 (Kovács, Liptai, Oajos, 2010. If a 2 4ab 4b 2 = 1, then there is no perfect power (a, b-baancing number. Remark. One can easiy check that the equation a 2 4ab 4b 2 = 1 has innitey many soutions in integers a, b with a > 0, b 0. Hence Theorem 6 competey soves the proposed probem for innitey many pairs (a, b. The foowing theorem takes up the probem where the poynomia f( in (9 has some combinatoria meaning. More precisey, we investigate binomia coecients ( k, products of consecutive integers Πk (, power sums S k ( and aternating power sums R k (. Note that the coecients of ( k, Sk ( and R k ( are not integers. Further, in the case f( = Π k ( Theorem 5 yieds a niteness resut, however, ony for the odd vaues of the parameter a. For these combinatoria choices of f( our net statement yieds a bound for the soutions of (9, without any assumptions for the parameters a and b. Theorem 7 (Kovács, Liptai, Oajos, 2010. Let k 2 and f( be one of the poynomias ( k, Πk (, S k 1 (, R k (. Then the soutions of equation (9 satisfy ma(m, < c 1 (a, b, k, where c 1 (a, b, k is an eectivey computabe constant depending ony on a, b and k. 15
In our na resut, under the assumption a 2 4ab 4b 2 = 1, we provide the compete soution of (9 with the above choices of f(, for some sma vaues of the parameter k. More precisey, we consider a cases where (9 can be reduced to an equation of genus 1. Further, we aso sove a particuar case of (9 which can be reduced to the resoution of a genus 2 equation. Theorem 8 (Kovács, Liptai, Oajos, 2010. Suppose that a 2 4ab 4b 2 = 1. Let f( { ( ( 2, ( 3, 4, Π2 (, Π 3 (, Π 4 (, S 1 (, S 2 (, S 3 (, S 5 (}. Then the soutions (m, of equation (9 are those contained in Tabe 6. For the corresponding parameter vaues we have (a, b = (1, 0 in a cases. f( ( Soutions (m, of (9 ( 2 (1, 3, (1, 4 ( 3 (2, 5, (2, 7 4 (2, 4, (2, 7 Π 2 ( (1, 3, (1, 2 Π 3 ( (1, 3, (1, 1 Π 4 ( S 1 ( (1, 4, (1, 3 S 2 ( (3, 8, (3, 9, (5, 27, (5, 28 S 3 ( S 5 ( Tabe 6: Soutions of equation (9 with the particuar poynomias Remark. We considered some other reated equations that ead to genus 2 equations. However, because of certain technica dicuties, we coud not sove them by the Chabauty method. We checked that under the assumption a 2 4ab 4b 2 = 1 equation (9 has no "sma" soutions (i.e. soutions with 10000 in cases f( { ( ( 6, 8, Π6 (, Π 8 (, S 7 (}. The resut of the fourth chapter are pubished in [27]. Amost fth powers in arithmetic progression A cassica theorem of Erd s and Sefridge [12] states that the product of consecutive positive integers is never a perfect power. A natura generaization is the Diophantine equation ( + d... ( + (k 1d = by n (10 16