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

or p = ±\ (mod 10); furthermore, if p * 5 and there are FPR's (mod p) then the number of FPR's is two or one, according to whether p = 1 (mod

4) or p = ~\ (mod

4). D. Shanks and L. Taylor (Fibonacci Quart 11. 1973, 159—160) have shown that if g is a FPR (mod p) then g1 is a FPR (mod p). M. J. DeLeon (Fibonacci Quart 15i 1977, 353—355) proved that there is a FPR (mod p) if and only M/(p) = p1. In [1] with P. Kiss we studied the connection between the rank of apparition of a prime p and the existence of FPR's (mod p). We proved that there is exactly one FPR (mod p) if and only if f{p) - P~ 1 o

r P = moreover, if p = 1 (mod 10) and there exitst two FPR's (mod p) or non FPR exists, then f(p)<p~ 1. M. E. Mays (Fibonacci Quart, 20. 1982, 111) showed that if booth p = 60k-\ and c/

= 30^-1 are primes then there is a FPR (mod p)

. In [16] we given some connections among the rank of apparition of p

e in the Lucas sequence R, the period of R modulo p

e, and Lucas primitive roots (mod p

e

); furthermore we shown necessary and sufficient conditions for the existence of Lucas primitive roots (mod p

e)

. Theorem 1.7. ([16]) Let R be Lucas sequence defined by integers A* 0 and B = - \, let p be an odd prime with D = A

2+44 0(modp), and let e>0 be an integer. Then there is a Lucas primitive root (mod p

e) if and only if *(p

e) = 4>(p

e) 123