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 On the other hand CQ

S (2tj+i

) 2 j

k k 2ntjk , ^ > 4 = > cos —cot > sin —. 13 Z S TT

« / J »] / J 77 v l<j<m n 1 < .7 < m " ' 1<J<™ 1 <t,k<n-l 1<í ,fc<n-1 !<<,/>< n - 1 The following hold true E

'Intik nk sin —cot— = 0, (14) n n l<j<m l<f,fc<n-l E

'lwtjk Txk . „. cos — cot — = 0 (15) n n 1 <j<m l<i,fc<n-l and V sin—— = 0. (16) 1' r? l<j<m Finally, by (11) together with (12), (13), (14), (15) and (16) we obtain: y- C°

A-(1 -C

A"

) y- r

o., 2trtjk Ak(n — 1) I Sk _ 9 h n l<j<m s is- l < j < m 1 < f, >c < n - 1 1 < f, is < n - 1 It is now clear 1 hat Theorem 3.2 and Lemma 3.3 imply the following Theorem 3.4. Let n he a natural number with n > 1 and m = [y/n ]. Then n is prime if and only if 2 7it jk ST *'

lLJ

K / i\ > cos = — m\n — 1). < J n 1 <]<m l<t,k<n- 1 References [1] APOSTOL

, T. M., The Resultant of the Cyclotomic Polynomials F

m(ax) and F

n{bx), Math. Comp. 29 (1975), 1-6. [2] DICKSON, L. E., History of the Theory of Numbers, vol. 1 (reprint), Chelsea, New York, 1952. [3] DIXON, J. D., Factorization and Primality Tests, Amer. Math. Monthly 91 (1984), 333-352. [4] DUDLEY, U., History of a. Formula for Primes, Amer. Math. Monthly 76 (1969), 23-28.