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

Acta Acad. Paed. Agriensis, Sectio Mathematicae 28 (2001) 3-11 ON PRODUCTS AND SUMS OF THE TERMS OF LINEAR RECURRENCES Peter Kiss &; Ferenc Mátyás (Eger, Hungary) Abstract. For a fixed integer m>

2, let }°°_

0 (l<i<m) be linear recursive sequences of integers, n

xi >X

2,...

|X

m =G^ G^ -G^ and let x — max (i,). In the paper it is proved, under some restrictions, that there are effectively computable constants c and n

0 such that |s — n

Tl it

J |> e

c x if s is an integer having fixed prime factors only, x>n

0 and Xj>j-x for any l<j<m with a fixed real number 0<7<1. Similar result can be obtained if we replace the pruduct of the terms by their sum. AMS Classification Number: 11B39, 11J86. Keywords: linear recursive sequence, linear forms in logarithms. 1. Introduction f ( '

i1 0

0 Let the linear recurrences G^

1' = <1 Gn r (2 — 1,2,..., m; m > 2) of order I J n0 ki be defined by the recursion (1) 6f = .4 f G^ + 4>G$L

3 +... + A$C%l

t

i (n > ki > 2), where the initial values G^p and the coefficients (j = 0,1,..., ki — 1) are rational integers. Denote the distinct roots of the characteristic polynomial (2) gW(z) = x

k> - A^x

k' of the sequence G (i

) defined in (1) by a 1 ,... ,a

t {ti > 2), and suppose that 4? (+ G i(0 • 1 + • • Research supported by the Hungarian OTKA Foundation, No. T 032898 and T 29330.