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
20 Péter Kiss and Béla Zay x-\-z Y Y p A.— A, "I- T\,Z>\ — A d<z and x+z ^) = ££ £ ~ *(*) = ££ £ d < z By Lemma 1 and 2, using (3) and the condition of the Theorem, we have *m = e(G + 0(I)) E p] £ ^ d<z \ r{p) = d ' ) r{p)<z y y F> (8) + ° £ J =« + 0«, \r(p)<2 / where c is a constant determined by Lemma 2. Now we give an estimation for B(x). Since every d with d > z occurs at most once in the sum, by Lemma 1 and z < x we get *<•>< £ E r £ r d<x+z r(p) — d r(p)<x+z = log log(x + z) + 0(1) = log log X + 0(1). But log log x = o(z) by the condition of the Theorem, so (9) B(x) = o{z). From (7), (8) and (9) the Theorem follows. Lastly we note that our theorem can be improved. In the proofs we have used only some elementary results of prime number theory. Using some deeper methods and results (e.g. the Brun-Titchmarsh inequality) the condition for z can be replaced by 2/i o g log log x —> oo.
