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

s are positive integers, then there exist infinitely many EulerLucas pseudoprimes with parameters A, B which are products of exactly s primes of the form ax+

1. A. Rotkiewicz (Bull. Acad. Polon. Sei. Ser. Sei. Math. Astr. Phys. 20, 1972, 349—354) gave a proper generalization of ordinary pseudoprimes for Lehmer sequences. Let U-U{L,M) be the non-degenerate Lehmer sequence defined by integers L, M and by (1.2). Let V - V(L,M) = {V

nY

n„

0 be the sequence defined V

0 = 2 and by the relation V

n = U

2

n/U

n '(/I = 1,2,... ). Similarly to the congruences (2.4)-(2.7), it is also known that for odd prime n with (», LK) = 1, we have (2.8) U

H -(LK/n) = 0 (mod«), (2.9) U„=(K/n) (mod n) (2.10) V

n = (L/n) (mod n), and for odd prime n with («, LK) - 1 (2.11) U{n

<LKin))i2(

) (mod zi) when (LM!n) = 1 V(n-(LK'n)),2 =

0 (mod w) when (LM/«) = 1. An odd composite n is called a Lehmer pseudoprime with parameters L, MM (n,LMK

) = 1 and (2.8) holds, and it is an Euler-Lehmer pseudoprime if (2.11) is true. Some results of 129