Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 2004. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 31)

Tsangaris, P. G., Prime numbers and cyclotomy

6 P. G. Tsangaris Then x

k — 1 is a divisor of K

k(x), and (1 - C*)1 = L

k

n(C)/Fn/d(l)*

{nm

n'

d\ where L

k(x) = K

k(x)/(x

k - 1). Proof. Immediate by using Theorems 1.1 and 1.2. Corollary 1.4. If n is a prime and k < n, then we have (1 _£*)-!= I Wf(n-w-l)

m Kw<n— 1 Proof. Here (n, k) = 1 and F

n( 1) = n, so by Theorem 1.3 we have L

k

n(x) = (F

n(x

k) - F

n(l))/(x

k - 1) = ^ 1 <w<n—

1 which proves the corollary. 2. A Primality Criterion The known formula of Hacks [5, p. 205] for the g.c.d. of two natural numbers (n, j) = 2 ^ [ji/n] - jn+j + n l<i<n-l together with the fact that n is prime if and only if ^^ (n, _/) = m where m = 1 < j <m [>/n] implies the following: Theorem 2.1. Let n be a natural number with n > 1, m = [>/" ] and g(n) = 4 [jí'/n] - (m - l)m(n - 1). l<J<m l<t<n-l Then the following hold true: (i

) n is prime if and only if y(n) — 0. (ii) ix is composite if and only if g(n) > 0.