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

where <D denotes the Euler function. There is exactly one Lucas primitive root (mod/?

6) if t(p

e) = $>(p

e) and p = -\ (mod 4), and there are exactly two Lucas primitive roots (mod p

e) if t(p

e) = $>(p

e) and p = 1 (mod 4). Theorem 1.8. ([16]) Let R be Lucas sequence deßned by integers A* 0 and B - -1, let p be an odd prime with D = A

2+4é 0(modp), and let e>0 be an integer. Then there is exactly one Lucas primitive root (mod p

e

) if and only if r(p

e) = <&(p

e) and /? = l(mod4), and exactly two Lucas primitive roots (mod p

e) exist if and only if r(p

e) = <$>{p

e)/2 and p si (mod 8) or r(p

e) = $>(p

e)/ 4 and /? = 5(mod8). From these theorems, some other results follow. Collaiy 1.9. If R, p and e satisfy the conditions of Theorem 1.8 and r(p

e) = 0(/?

e), then g is a Lucas primitive root (mod p

e) if and only if x = g satisßes the congruence R

nx + R

n_^-\(modp

e), where n - O(p

e)/2. Corollary 1.10. If R, p and e satisfy the conditions of Theorem 1.8 and g is a Lucas primitive root (mod p

e ), then g-A is a primitive root (mod p

e). 124