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

R

n+

] = AR

n-BR

n_

l for n>0. The sequence i?(l,-l) is the Fibonacci sequence F. Let p be an odd prime with Bé 0 (mod p) and let e > 0 be an integer. The positive integer r = r(p

e) is called the rank of apparition of p

e in the sequence R if R

r = 0 (mod p

e

) and R

má 0 (mod/?*) for 0<m <r

; furthermore w(p

e) is called the period of the sequence R modulo p

e if it is the smallest positive integer for which R

n= 0 (mod//) and = i(mod/?

e). In the Fibonacci sequence, we denote the rank of apparition of p

e and period of F modulo p

e by / (p

e

) and/(p

e), respectively. Let the number R be a primitive root (mod/?

6). If x = g satisfies the congruence (1.7) f(x) = x

2-Ax+B = 0(mod p

e), then we say that R is a Lucas primitive root (mod p

e

) with parameters A and B

. This is the generalization of the definition of Fibonacci primitive roots (FPR) modulo p that was given by D. Shanks for the case A = -B = 1 (Fibonacci Quart,

, 10.1973,163—168,181). The conditions for the existence of FPR (mod p) and their properties were studied by several authors. For example, D. Shanks proved that if there exists a FPR (mod p

e) then p = 5 122