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

Theorem 2.5 ([9]) Let c(< 1) and k be fixed positive integers. Then there are infinitely many composite integers n satisfying the congruence (2.12). In [17] we considered the following congruence (2.13) a

n'

k = b

n

k (mod«), where a,b and k are given positive integers with conditon (a, b) = 1. Improving Theorem 2.5 we proved the following Theorem 2.6. ([17]) The congruence (2.13) has inßnitely many composite solutions n if neither (a,b,k

) is one of the following triples: (2

U +1,2" -1,3) for u>\, (5.2

V + 1,5.2

V -1,3) for v>0, (c +

1, c, 2); (c + 3, c, 2) forol. We note that W. L. McDaniel (Colloq. Math. 59, 1990, 177—190) independently proven this theorem and some generalizations of it We obtained a similar result of Theorem 2.6 for Lehmer pseudoprimes. Theorem 2.7. ([17]) Let U = U(L,M) be a non-degenerate Lehmer sequence. Then there is a positive integer k

0 such that for any fixed k > k

0 the congruence 133