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

K«-(d/«))/2 =

0 (mod«) when (B/n) = l P(«-(dm))/2 = 0 (mod n) when (2? / w) = -1, where (In) is the Jacobi symbol. If n is composite, (n,2D) = \, but (2.4) still holds, then n is called a Lucas pseudoprime with parameteres A

fB. Furthermore, if n is composite, (n,2D) = \ and satisfies the congruence (2.7), then n is called Euler-Lucas pseudoprime. It can be easily seen that in the case when A = c +1 and B = c, by using (1.4), we have R

n = (c

n - l)/(c1), and so the definitions of Lucas and Euler-Lucas pseudoprimes are generalizations of pseudoprimes and Euler pseudoprimes to base c > 1. We list some results which are in connection with ours. C. Pomerance, J. L. Selfridge and S. S. Wagstaff, Jr. (Math. Comp. 35, 1980, 1003—1026) proved that for given positive integer 5 there is an Euler-pseudoprime which is a product of exactly 5 distinct primes. From results of A. J. van der Poorten and A. Rotkiewicz (J. Austra. Math. Soc. Ser. A 29, 1980, 316—321) it follows that there are infinitely many Euler pseudoprimes to base an integer c> 1, which are of the form ax+b

, where (a,b) = 1. On the other hand, P. Erdős (Amer. Math

. Monthly 56, 1949, 623—624) and E. Iieuwens (Doctor thesis, Delft, 1971) proved that for any integers c

, s>\ there are infinitely many pseudoprimes to base c which are products of exactly s primes. This result was extended by P. lűss, B. M. Phong and E. Iieuwens in [5] for Euler-Lucas pseudoprimes, among others, we proved that if R - R(A,B) is a non-degenerate Lucas sequence with D - A

2 - 4B > 0 and a

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