Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1995-1996. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 23)

KlSS, P. and ZAY, B., A note on the prime divisors of Lucas numbers

A note on the prime divisors of Lucas numbers PÉTER KISS* and BÉLA ZAY* Abstract. A sequence R

0, R

I

T R

2,--- of rational integers is called sequence of Lucas numbers with parameters A and B if R

N=AR

n-i+BR

N2 (N>l) and the initial terms are H

o=0, RI=L. Let /(n) be the reciprocal sum of the distinct prime divisors of R

N. It is known that there is a constant c>0 such that ^ f(n) = cx+0(x). We show that the n < x average order of f(n) is also aconstant if we consider the function only on a short interval [x,x+z], where 2/log log x-+oo if X—+00. Let £

0, be a sequence of Lucas numbers with parameters A and B defined by R

n = AR

n_ 1 + BR

n_2 (n > 1), where A, B are fixed nonzero coprime rational integers and the initial terms are RQ = 0, R\ = 1. Denote by a and ß the roots of the equation x

2 — Ax — B — 0. In the following we suppose that the sequence is a non degenerate one, i.e. <y/ß is not a root of unity. It is known that the terms of this sequence can be expressed by a

n - 3

n (1) Rn = aß for any n > 0. It is also known that if p is a prime and p\B, then there are terms of the sequence divisible by p. We denote the least positive index of these terms by r(p). Thus p | i?

r(

p) but p\R

m for 0 < m < r(p). If p\B, D = A

2 + 4B and ( D/

P

) denote the Legendre symbol with (D/

P

) = 0 in the case p I D, then we have (2) r(p)l(p-(%)) and (3) p I R

n if and only if r(p) \ n (see e.g. D. H. Lehmer [4]). * Research supported by the Hungarian OTKA foundation, N° T 016975 and 020295.