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

there exist primes q and r such that pqr is a super pseudoprime to base 2. In [6] we extended the result of Rotkiewicz and the result of Fehér and Kiss mentioned above proving that form every integers a > 1 and c > 1 there are infinitely many triplets of distinct primes p

, q and r of the for ax +1 such that pqr is a super pseudoprime to base c

. We also showed that if the square-free kernel of the base c is congruent to ±1 modulo 4, then the series Zl/logn is divergent, where n runs through all super pseudoprimes to base c which are products of exactly three distinct primes. P. Kiss (Ann. Univ. Sei. Budapest Eötvös Sect Math. 28, 1986, 153—159) studied the super Lucas pseudoprimes for non-degenerate Lucas sequences R(A,B

) and proved that R

2

p / A is a super pseudoprime with parameters A,B for every large prime p, furthermore he showed that the series XI/log n, where n runs through all super Lucas pseudoprimes with parameters A and B

, is divergent In [11], by using some result of J. Wójcik (Acta Arith. 40, 1981/82, 155—174; 41, 1982, 117—131) we improved above result as follows: Theorem 2.8. ([11]) LetU = U(L,M) be a non-degenerate Lehmer sequence. Then there exists a positive integer w, such that for infinitely many primes p of the form ax + b

, where (a,b) = 1 and b = 1 (mod(a,

>*>,)), there are primes q and r such that pqr is a super Lehmer pseudoprime with 135