Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1993. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 21)

Bui Minh Phong: Recurrence sequences and pseudoprimes

parameteres L,M. The constant w

x is effectively computable in the terms of L and M

. Theorem 2.9. ([11]) LetU - U(L,M) be a non-degenerate Lehmer sequence with condition LK - L(L - AM) > 0 and let a> 1 be an integer. Then there are infinitely many triplets of distinct primes p,q and r of the form ax+

1 such that pqr is a super Lehmer pseudoprime with parameteres L,M. Theorem 2.10. ([11]) Let U = U(L,M) be a nondegenerate Lehmer sequence. Let S

x and S

2 denote the set of all super Lehmer pseudoprimes with parameters L,M which are determined in Theorem 2.8 and Theorem 2.9, respectively. Then the series y _L and y -LneSi log" n

es, are divergent We note that the conditions of Theorem 2.8 are satisfied for any integer a> 1 if 6 = 1 and for every pairs a,b with (

ű

,^,^) = 1. it is obvious that these results remain valid if we replace the super Lehmer pseudoprimes with super Lucas pseudoprimes. For example, from Theorem 2.10 we get Corollary 2.11. For every integers a,c>\ the series El/log», where n runs through all super pseudoprimes to base c which are products of exactly three distrinct primes of the form ax+\,is divergent 136