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

many composite integers n such that n\(c" k -1)?" (Pseudoprime Numbers and Their Generalizations

, Univ. of Novi Sad\ 1972, problem 18). It is known as above that the answer is affirmative in the case k = 1; the numbers satisfying the condition are pseudoprimes to base c

. A general result was obtained by A Makowski (Simon Stevin 36, 1972, 71): For any natural number k> 2 there are infinitely many composite n such that (2.12) 1 (mod«) for any positive integer c with (c,w)=l. This result was proved earlier by D. C. Morrow (Amer. Math. Monthly 58, 1951, 329—330) in the case k = 3. In this proof, Makowski showed that there are infinitely many integers n of the form n-p.k (where p is a prime) such that congruence (2.12) holds for any positive integer c if (c,w)=l. Naturally, (k

, c) - 1 for these numbers, and so the question remained unanswered if c and k are fixed and (£,c)>l. In the case (k,c)>\, A. Rotkiewicz obtained two results: He proved that (2.12) has infinitely many solutions if k-3 and c is an arbitrarily fixed positive integer, or if k-2 and c- 2 (see Theorem 32 in his book and Math. Comp. 43, 1984, 271—272, respectively)

. In [9] with P. Kiss we gave a general solution of the problem, namely we proved 132