Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 2001. Sectio Mathematicae (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 28)

Kiss, P. & MÁTYÁS, F., On products and sums of the terms of linear recurrences

6 P. Kiss & F. Mátyás Corollary. Under the conditions of Theorem 1, 11^ j,

2r

i,

iI

m ^ S if x = max (x

;) > 1 < i < m n

0. Theorem 2. Let G^ (1 < i < m, 2 < m) be sequences defined by (3) if 2 < i < ??? and by (4) if i = 1 and let S be the set of integers defined by (5). Suppose that G^ is the dominant sequence (with the dominant root c*i = a^

1'

) among the sequences G^' (1 < i < rn). Then there exist positive real numbers c\ and n\ such that if (11a and 116) ciicx^

1 £ S and x\ > max (xj), 2 <i<m then I

s ~ ,IS l

m I > e 1 1 for any s G S and positive integers x

i

} x

2, • • •, x

m satisfying the condition Xi > n\. The constants c\ and ni are effectively computable positive numbers depending only on the primes pi,p2, • • • ,Pr and the parameters of the sequences G^' (1 < i < m). Corollary. Under the conditions of Theorem 2, _ S if xi > nj.. 3. Lemmas and Proofs To prove the theorems we need the following auxiliary results. Lemma 1. Let A = 7o + 7i • logujx + 72 • logu

2 + b J

n • logui

n

, where the 7

's and ui's denote algebraic numbers (u>i / 0 or 1). We assume that not all the 7's are zero and that the logarithms mean their principal values. Suppose that uii and 7,; have heights at most M{

(> 4) and D(> 4), respectively, and that the field generated by the ui's and 7

's over the rational numbers has degree at most d. If A / 0, then |A| > where Q = log Mi • log M

2 • • • log M

n

, ÍÍ' = ft/ log M

n and C = (16ud)

200

n. If 70 = 0 and 71,72, • • •, 7

n ar

c rational integers, then |A| > B~

Cillo&f

l'