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

T

0(k) :=k, T

n(k): = R^ /R

n (n=l,2,...). As it is known, T

n(k) - s are integers. Let T(k) = [T

n(k)} . L. Somer (Fibonacci Quart 22, 1984, 98—100) proved that the sequence T(k) is a linear integeral recurrence of order k

, furthermore the order k is minimal. Indeed, by using (1.4) we get where a. = /"'^1. If D = A

2-4B <0, then |aj=... =1^,1=1 rl*"

1Consequently, the investigation of the Diophantine equation T

x = sy

q has meaning. In [12] we proved with I. Joó that the Diophantine equation T

x(k) = sy

q in integers SGS, q>2, x, |_y|> 1 implies max(|.y|,|_y|,jc,^) < C

5, where C

5 is an effectively computable constant depending only on A, B, k and 5. By using the theorem of T. N. Shorey, A van der Poorten, R. Tijdeman and A Schinkel (Transcendence Theory, New York, 1977) concerning the Thue-Mahler equation and the theorem of C. L. Stewart (Transcendence Theory, New York, 1977) on linear forms in logarithms of algebraic numbers, in [10] (Theorem 3.1) we improved the above result, namely we showed the following 120