A note on the products of the terms of linear recurrences

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1 Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae, 24 (1997) pp A note on the products of the terms of linear recurrences LÁSZLÓ SZALAY Abstract For an integer ν>1 let G (i) (,,ν) be linear recurrences defined by In the paper we show that the equation G (i) n =A(i) 1 G(i) n 1 + +A(i) k i G n ki (n k i ) dg (1) x 1 G (ν) xν =swq, where d,s,w,q,x i are positive integers satisfying some conditions, implies the inequality q<q 0 with some effectively computable constant q 0 This result generalizes some earlier results of Kiss, Pethő, Shorey and Stewart 1 Introduction Let G (i) = {G (i) n } n=0 (i = 1,2,,ν) be linear recurrences of order k i (k i 2) defined by (1) G (i) n = A (i) 1 G(i) n A(i) k i G (i) n k i (n k i ), where the initial values G (i) j (j = 0,1,,k i 1) and the coefficients A (i) l (l = 1,2,,k i ) of the sequences are rational integers We suppose, that A (i) k i 0 and there is at least one non-zero initial value for any recurrences By α (i) 1 = γ i,α (i) 2,,α(i) t i we denote the distinct roots of the characteristic polynomial p i (x) = x k i A (i) 1 xk i 1 A (i) k i of the sequence G (i), and we assume that t i > 1 and γ i > α (i) j for j > 1 Consequently γ i > 1 Suppose that the multiplicity of the roots γ i are 1 Then the terms of the sequences G (i) (i = 1,2,,ν) can be written in the form (2) G (i) n = a i γ n i + p (i) 2 (n) ( α (i) 2 ) n + + p (i) t i (n) ( α (i) t i ) n (n 0),

2 48 László Szalay where a i 0 are fixed numbers and p (i) j (j = 1,2,t i ) are polynomials of (see eg [8]) Q(γ i,α (i) 2,,α(i) t i )[x] A Pethő [4,5,6], T N Shorey and C L Stewart [7] showed that a sequence G(= G (i) ) does not contain q-th powers if q is large enough Similar result was obtained by P Kiss in [2] In [3] we investigated the equation (3) G x H y = w q where G and H are linear recurrences satisfying some condititons, and showed that if x and y are not too far from each other then q is (effectively computable) upper bounded: q < q 0 2 Theorem Now we shall investigate the generalization of equation (3) Let d Z be a fixed non-zero rational integer, and let p 1,,p t be given rational primes Denote by S the set of all rational integers composed of p 1,,p t : (4) S = {s Z: s = ±p e 1 1 pe t t, e i N} In particular 1 S (e 1 = = e t = 0) Let (5) G(x 1,,x ν ) = G (1) x 1 G (ν) x ν be a function defined on the set N ν By the definitions of the sequences G (i) s G takes integer values With a given d let us consider the equation dg(x 1,,x ν ) = sw q in positive integers w > 1, q, x i (i = 1,2,,ν) and s S We will show under some conditions for G that q < q 0 is also fulfilled if q satisfies the equation above Exactly, using the Baker-method, we will prove the following Theorem Let G(x 1,,x ν ) be the function defined in (5) Futher let 0 d Z be a fixed integer, and let δ be a real number with 0 < δ < 1 Assume that G(x 1,,x ν ) ν a i γ x i i if x i > n 0 (i = 1,2,,ν) Then the equation (6) dg(x 1,,x ν ) = sw q

3 A note on the products of the terms of linear recurrences 49 in positive integers w > 1, q, x 1,,x ν and s S for which x j > δ max i {x i } (j = 1,2,,ν), implies that q < q 0, where q 0 is an effectively computable number depending on n 0,δ,G (1),,G (ν) 3 Lemmas In the proof of our Theorem we need a result due to A Baker [1] Lemma 1 Let π 1,π 2,,π r be non-zero algebraic numbers of heights not exceeding M 1,M 2,,M r respectively (M r 4) Further let b 1,b 2,, b r 1 be rational integers with absolute values at most B and let b r be a non-zero rational integer with absolute value at most B (B 3) Suppose, that r b i log π i 0 Then there exists an effectively computable constant C = C(r,M 1,,M r 1,π 1,,π r ) such that r (7) b i log π i > e C(log M r log B + B ) B, where logarithms have their principal values We need the following auxiliary result Lemma 2 Let c 1,,c k be positive real numbers and 0 < δ < 1 be an arbitrary real number Further let x 1,,x k be natural numbers with maximum value x m = max i {x i } (m {1,,k}) If x j > δx m (j = 1,,k) and x m > x 0 then there exists a real number c > 0, which depends on k,δ,max i {c i } and x 0, for which (8) e c ix i < e c(x 1+ +x k ) = e cx, where x = x x k Proof of Lemma 2 Using the conditions of the lemma we have e c ix i < e c iδx m = where d i = δc i If d m = min i {d i } then e d ix m, e d ix m ke d mx m = e log k d mx m

4 50 László Szalay Since x m x 0, it follows that e log k d mx m e d m x m = e ckx m e cx with a suitable constant d m and c = d m k 4 Proof of the Theorem By c 1,c 2, we denote positive real numbers which are effectively computable We may assert, without loss of generality, that the terms of the recurrences G (i) are positive, d > 0, s > 0 and the inequality (9) γ 1 γ 2 γ ν also holds Let us observe that it is sufficient to consider the case x i > n 0 (i = 1,2,,ν) Otherwise, if we suppose that some x j n 0 (j {1,2,,ν}) then x m = max i {x i } cannot be arbitrary large because of the assertion x j > δx m It means that we have finitely many possibilities to choose the ν-tuples (x 1,,x ν ), and the range of G(x 1,,x ν ) is finite So with a fixed d, if inequality (6) is satisfied then q must be bounded In the sequel we suppose that x i > n 0 (i = 1,2,,ν) Let x 1,, x ν, w, q and s S be integers satisfying (6) We may assume that if (10) s = p e 1 1 pe t t then e j < q, else a part of s can be joined to w q Using (2), from (6) we have ( ( ) (11) sw q = d a i (γ i ) x i 1 + p(i) 2 (x i) α (i) xi 2 + ) a i γ i A consequence of the assumptions γ i > α (i) j (1 < j t i) is that ( ( ) (12) 1 + p(i) 2 (x i) α (i) xi 2 + ) 1 whenever x i a i γ i Hence there exist real constants 0 < ε 1,,ε ν < 1 such that d a i γ i x i (1 ε i ) < sw q < d a i γ i x i (1 + ε i ),

5 and A note on the products of the terms of linear recurrences 51 c 1 γ i x i < sw q < c 2 γ i x i As before, let x = x x ν and applying (9) we may write log c 1 + xlog γ ν < log s + q log w < log c 2 + xlog γ 1 Since log s 0, we have (13) log c 3 + xlog γ ν < q log w < log c 2 + xlog γ 1 with c 3 = c 1 s From (13) it follows that x (14) c 4 q < log w < c x 5 q with some positive constants c 4, c 5 Ordering the equality (11) and taking logarithms, by the definition of ε i we obtain ( ) Q = log sw q d ν a i γ i x i = log 1 + p(i) 2 (x i) α (i) xi 2 + a i γ i < < ν log 1 + ε i ν e c i x i, where Q 0 if we assume, that x i > n 0 for every i = 1,2,,ν, and c i is a suitable positive constant (i = 1, 2,, ν) Applying Lemma 2 and using the notation x = x x ν, it yields that (15) Q < e c 6(x 1 + +x ν ) = e c 6x On the other hand (16) Q = log s+q log w log d log a i x 1 log γ 1 x ν log γ ν, where log s = e 1 log p e t log p t (see (10)) Now we may use Lemma 1 with π r = w = M r, since the ordinary heights of p j (j = 1,2,,t), d, ν a i and γ i (i = 1,2,,ν) are constants So B = q In comparison

6 52 László Szalay the absolute values of the integer coefficients of the logarithms in (16), we can choose B as B = x So by (16) and Lemma 1 it follows that (17) Q > e c 7(log w log q+ x q) Combining (15) and (17) it yields the following inequality: ( (18) c 6 x < c 7 log w log q + x ), q and by (14) it follows that (19) c 6 x < c 7 (log w log q + 1 ) log w < c 8 log w log q c 4 with some c 8 > 0 Applying (14) again, we conclude that 1 c 5 q log w < x and so by (19) (20) c 9 q < log q follows But (20) implies that q < q 0, which proves the theorem References [1] A Baker, A sharpening of the bounds for linear forms in logarithms II, Acta Arith 24 (1973), [2] P Kiss, Pure powers and power classes in the recurrence sequences, Math Slovaca 44 (1994), No 5, [3] K Liptai, L Szalay, On products of the terms of linear recurrences, to appear [4] A Pethő, Perfect powers in second order linear recurrences, J Num Theory 15 (1982), 5 13 [5] A Pethő, Perfect powers in second order linear recurrences, Topics in Classical Number Theory, Proceedings of the Conference in Budapest 1981, Colloq Math Soc János Bolyai 34, North Holland, Amsterdam,

7 A note on the products of the terms of linear recurrences 53 [6] A Pethő, On the solution of the diophantine equation G n = p z, Proceedings of EUROCAL 85, Linz, Lecture Notes in Computer Science 204, Springer-Verlag, Berlin, [7] T N Shorey, C L Stewart, On the Diophantine equation ax 2t +bx t y+ cy 2 = d and pure powers in recurrence sequences, Math Scand 52 (1987), [8] T N Shorey, R Tijdeman, Exponential diophantine equations, Cambridge, 1986 László Szalay University of Sopron Institute of Mathematics Sopron, Ady u 5 H-9400, Hungary laszalay@efehu