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

the congruences (2.4), (2.5) and (2.6) simultaneously. A similar result also holds for Lehmer sequences. • 's Theorem 2.3. ([7]) Let (J -U{L,M) be a non-degenerate Lehmer sequence for which M = ±1 and LK - L(L± 4) > 0. Then for any fixed positive integer s there are infinitely many Euler-Lehmer pseudoprimes n which are products of exactly s distinct primes of the form ax+

1 and satisfy the congruences (2.8), (2.9) and (2.10) simultaneously. In the following we say that n is a perfect Lehmer pseudoprime with parameteres L,M if (

n,2LMK

) = 1 and the congruences (2.8), (2.9), (2.10) hold. Improving a result of [8] concerning Lucas sequences, in [10] we showed Theorem 2.4. ([10]) Let U = U(L, M) be a non-degenerate Lehmer sequence. Then the following three conditions are dependent (i) n is a perfect Lehmer pseudoprime with parameters L,M (ii) n is an Euler-Lehmer pseudoprime with parameters L,M (iii) n is an Euler pseudoprime to base M. That is, from any two ones of them, the third one follows. II.2. A generalized solution of A. Rotkiewicz's problem A. Rotkiewicz asked in his book the following question. "Let c,k > 1 be fixed positive integers. Do there exist infinitely 131