Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1997. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 24)

SZALAY, L., A note on the products of the terms of linear recurrences

48 László Szalay where ai ^ 0 are fixed numbers and p^ (j = 1,2..., £;) are polynomials of QtTi,"!

0,...,^)!«] (see e.g. [8]). A. Pethö [4,5,6], T. N. Shorey and C. L. Stewart [7] showed that a sequence G(= G^

) does not contain g-th powers if q is large enough. Similar result was obtained by P. Kiss in [2]. In [3] we investigated the equation (3) G

xH

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^. 2. Theorem Now we shall investigate the generalization of equation (3). Let d E Z be a fixed non-zero rational integer, and let pi,...,p

t be given rational primes. Denote by S the set of all rational integers composed of pi,

... ,p

t: (4) 5 = {s E Z : 5 = ±p[> • • -pj'

, a E N} . In particular 1 E S (ei = • • • = e

t = 0). Let (5) G(x

u...,xJ = G<»...G%> be a function defined on the set N". By the definitions of the sequences Q takes integer values. With a given d let us consider the equation dQ(x 1,... ,x

y) = sw

q ,in positive integers w > 1, q, X{ (i = 1,2,.. and 5 E S. We will show under some conditions for Q that q < qo is also fulfilled if q satisfies the equation above. Exactly, using the Baker-method, we will prove the following Theorem. Let Q[x\

,..., x„) be the function defined in (5). Father let 0 d E Z be a fixed integer, and let 6 be a real number with 0 < S < 1. Assume that G(xi

,..., x„) / fj a^f

' if X{ > tlq (z = 1, 2,..., v). Then the i=

1 equation (6) dQ(x 1,..., x„) = sw

q