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

Prime numbers and cyclotomy 9 3. Prime numbers, roots of unity, cyclotomy and trigonometry By Volosin's Theorem [10] we have: a n — 1 1 v n I _ ÍS K ' n 2 n n ' ] — i Ks<n-1 s for any pair of (positive) integers a,n. Hence by (5) and Theorem 2.1 we have the following: Theorem 3.1. Let n be a natural number with n > 1 and m = [>/"•]• Then, n is prime if and only if *k(tj+

1) . / "I'JTM E T37T = "•(»Theorem 3.2. Let n be a natural number with n > 1 and m = [\/n ]. Then n is prime if and only if 1 < t~k < n - 1 Proof. If n is a prime, by Theorem 3.1 and Corollary 1.4 we obtain: - ^ C

(tj + 1)

k J2 wC**"-""

1* = m(n - 1). (7) i<j<™ l<w<n— 1 1 < ( , k < n - 1 — Let (

k = l/z. Clearly (

k / 1, i.e. 1. Therefore E = ^ E «"'-IS- <»» \<w<n-\ " 1 < w < n — 1 ^ ^ By (7) and (8) follows (6). Assume now that (6) holds true. We have C

/

v'

(

n~

1

) + (

k -2^0 and ^ 1 because (

k ^ 1. Also, the following hold true: 1 - (

k 1 - 1) _ 2 -1) _ \ Hence l-£

f

c) +1) ^fc(n-l) _ 2 ~~ 1 _ '