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

holds only for finitely many sequences W and for finitely many integers x. KL Győry (Acta Arith. 40, 1982, 369—373) improved this result giving explicit upper bound for x and for the constants of Lucas sequences which satisfy (1.5). The Diophantine equation (1.6) W

x = sy

q was also studied by several authors. T. N. Shorey and C. L. Stewart (Math. Scand. 52, 1983, 24—36) proved that if y > 1, q > 1 are integers and W is a non-degenerate recurrence of order k for which w, = 1 and \a

x\>\aj\ (J = 2,...,/), then (1.6) implies the inequality q < C

4, where C

4 is an effectively computable constant in the terms of 5 and the parameters of sequence W. They showed that x and y are also bounded for second order recurrences. A. Pethő (J. of Number Theory 15, 1982, 5—13) proved similar results for second order recurrences supposing (^,4) = ! and seS. For recent general results we refer to the monograph by T. N. Shorey and R. Tijdeman (Exponential Diophantine Equations, Cambridge University Press, 1986), further to the references there. The following problem remained open : if |aj=. then the equation (1.6) has finite or infinite solutions? Let R = R(A,B) be a Lucas sequence defined by integers A,B. For fixed integer k > 0 we put 119