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

A note on the products of the terms of linear recurrences 49 in positive integers w > 1, q, X\

,..., x

u and s £ S for which Xj > ó max^a;;} (j = 1

,2,..., v), implies that q < q

0, where qo is an effectively computable number depending on no, 6, \ ..., G^

. 3. Lemmas In the proof of our Theorem we need a result due to A. Baker [1]. Lemma 1. Let , "k

2,

..., 7i> be non-zero algebraic numbers of heights not exceeding M\

, M

2,..., M

r respectively (M

r > 4). Further let b\

, b

2,..., 6

r_i 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 t>i log 7Tj / 0. Then there exists an effectively computable constant C = C(r, M\,..., M

r_ 1, 7Ti,..., 7r

r) such that (7) 22 bi logTT, >

e-C(lo

SM

rlo

EB' + f

r)

) t = 1 where logarithms have their principal values. We need the following auxiliary result. Lemma 2. Let ci,...,c

k be positive real numbers and 0 < S < 1 be an arbitrary real number. Further let xi,...,x

k be natural numbers with maximum value x

m = max,{x,} (m £ {!,...,&}). If Xj > Sx

m (j = 1,..., k) and x

m > xq then there exists a real number c > 0, which depends on k, ő, maxijc;} and xq, for which k e~CiXi ^ e-c(xi + --- + x

k) _ e~cx (8) £ i-1 where x = x\ + • • • + x

kProof of Lemma 2. Using the conditions of the lemma we have k k k Y^e~

c'

x' < Y^e0'

5*™ = J2

e~

d'

X

m, i=l z = l i=

1 where d{ = ÖC{. If d

m = min

t{</,•} then k ^^ e-djX

m ^ fc

e-d

mx

m __ glog k — d

mx

r t=l