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

the case where A--B1, the sequence /?(1,-1) is the Fibonacci sequence and we denote its terms by F

0

,F

x

, F

2.... D. H. Lehmer (Ann. Math. 31,1930,419—448) generalized some results of Lucas on the divisibility properties of Lucas numbers to the terms of the sequence U = U(L,M) = 0 which is defined by integer constants L,M,U

0=Q,U

] =1 and the recurrence where LM ^ 0 and K = L-4M ^0. The sequence U is called a Lehmer sequence and U

n is called a Lehmer number. We also say that the sequence U(L,M) is nondegenerate if at

ß is not root of unity, where a and ß denote the roots of z

2 -L

1/

2z + M = 0. It should be observed that Lucas numbers are also Lehmer numbers up to a possible multiplication by an integer factor. 1.1. Generalized Lehmer sequences In [18] we define a generalized Lehmer sequence as follows: Let H

Q

7H

x,L and M be integers with conditions IM * 0, K = L-4M

>0 and IJ^I+I/ZJ^Ö. A generalized Lehmer sequence is a sequence H

(

j,H

x,...H

n,... of integer numbers satisfying a relation (1.2) (mod 2) (mod 2), 112