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 19 Proof. By (2) r(p) < p + 1 and so (4) £ - > + 0(1)= log logy+ 0(1). r(p)<y ^ P<y On the other hand, by (1), (5) |Ä„| < exp(cin) with some c\ > 0. By (5) R

n has at most C2n distinct prime divisors since the product of the first c'2n primes is greater than e

Ci

n for some C2. From this it follows that the number of primes for which r(p) < y is less than c

3y

2. But each of the first c^y

2 primes is less than c

4y

3 and so (6) E -< E -= log logy+ 0(1). r(p)<y p<c

ty

3 From (4) and (6) the lemma follows. Lemma 2. For the primes with p\B the sum y — pr(p) P I B convergens

. Proof. Since p | r

r(

p), by (5) p < exp(c

1r(p)) and logp < cxr(p) follows. So 11 y —TT < c5 V 1 p P

r{P) p Piogp P\B It is known that the last sum is convergent which proves the lemma. Proof of the Theorem. Let x and z be sufficiently large positive integers. We can suppose that z < x since in the case z > x the Theorem follows from the case z < x. The sum in the Theorem can be written as x-\-z (7) £ f(n) = A(x) + B(x), n=x where