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

LIPTAI, K. and TÓMÁCS, T., Pure powers in recurrence sequences

Pure powers in recurrence sequences 37 The purpose of this paper to generalize this result. We show that the under certain conditions the number of the solutions of equation G

nG X

xG x

2 • • • G

X

kG

x = w

q where n is fixed, are finite. We use a well known result of Baker [1]. Lemma. Let 71,..., j

v be non-zero algebraic numbers. Let Mi

,.. ., M

v be upper bounds for the heights of 71,... ,7

v, respectively. We assume that M

v is at least 4. Further let b\,

... ,6^-1 be rational integers with absolute values at most B and let b

v be a non-zero rational integer with absolute value at most B'

. We assume that B' is at least three. Let L defined by L = bi log 7! + b Mog7„, where the logarithms are assumed to have their principal values. If L 0, then |Z| > exp(—C(log B' log M

v + B/B')), where C is an effectively computable positive number depending on only the numbers Mi,...,M

v_i, 71,...,~/

v and v (see Theorem 1 of [1] with = 1 IB'). Theorem. Let G be a k

i

h order linear recursive sequence satisfying the above conditions. Assume that a / 0 and G{ / aa

l for i > tlq. Then for any positive integer n

, k and K there exists a number qo, depending on n, G

, K and k, such that the equation (2) G

nG

X

lG

X

2 • • •

G

X

kG

x = w

q (n < xi < • • • < x

k < x) in positive integer X\

, x

2

,..., x

k, x, w, q has no solution with x

k < Iin and Q > QoProof of the theorem. We can assume, without loss of generality, that the terms of the sequence G are positive. We can also suppose that n > no and n sufficiently large since otherwise our result follows from [20] and [7]. Let xi,x2,

... ,xk,x,w,q positive integers satisfying (2) with the above conditions. Let e

m be defined by := ir

2(m)f^-)

m + ir

3(m)f^)

m + -..+ ir

s(m)f^)

m (m > 0). a \ a J a \ a / a \ a /