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

U„-k(LK/n) — 0 (mod ri) has infinitely many composite solutions n. Moreover, if k>\ and (k,M)= I, then there exist infinitely many composite integers n satisfying a congruence U

n_

k= 0 (mod«). II. 3. Super Lucas and super Lehmer pseudoprimes We say that n is a super pseudoprime to base integer c > 1 if each divisor of it is a prime or a pseudoprime to base c

. Similarly to super pseudoprimes to base c

, we say that n is a super Lucas (super Lehmer) pseudoprime if each divisor of it is a prime or a Lucas (Lehmer) pseudoprime. K. Szymiczek (Elem. Math. 21. 1966, 59) showed that F

nF

n+

l is a super pseudoprime to base 2 for any n > 1, where F

n=2

2 +1 is the «-th Fermat number. From the result of K. Szymiczek (Colloq. Math. 13, 1964/65. 259—263) it follows that there are infinitely many super pseudoprimes to base 2 which are products of exactly three primes. This result was extended by J. Fehér and P. Kiss (Ann. Univ. Sei. Budapest Eötvös Sect Math. 26, 1983.157—159) for super pseudoprimes to base c

, where c> 1 is an integer with c#0(mod4). A. Rotkiewicz (Glasgow Math. J. 9, 1968, 83—86) has obtained another generalization of Szymiczek's result, he proved that for infinitely many primes p of the form ax + b, where (a,b

) = 1, 134