Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1991. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 20)

Aleksander Grytczuk and Jaroslaw Grytczuk: On generators in multiplicative group of the Fieid Zp

- 41 p=8k+3 then -1 and the number 2 is a quadratic non—residue modulo p. From Lemma i we get that 2 is a generator in Z

p. On the other hand we have 2Cq+l) = 2q+2 = 2q+l+l = p+1 = lCmod p> and therefore q+1 is a generator in Z^ as the inverse element with respect to 2. Let p=8k+7, then 2 is not a generator in Z*. Since there are exactly £>Cp-l) generators in Z

p and the element 1 is not a generator thus there exists at least one >ii number g such that Cg,p—1)>1 and g is a generator in Z^. Because p=2q+l thus q|p-l and Cq,p-l)>l and 2 is not a # generator thus the number q must be a generator in Z

p. Ve have qC2q-l

> = qC2q+l-2) = qCp-2) s -2q = 1-p = lCmod p)

. tfi and so 2q—

1 is a generator in Z^. The last part of assertion follows from Lemma 2 and the proof is complete. REFERENCES 111 J.D. Baum, A note on primitive roots, Math,Mag. 38 C1965> 12-14. 1

2 Í A. Wilansky, Primitive roots without quadratic reciprocity, Math. Mag. 49 (1976>, 146.