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

8 P. G. Tsangaris Hence by our assumption we have: stjkn (k(tj+1) ^fc(n-l) _i_ £k _ O I — C

k 1 < j < m S 'S" l<j<m S 1<(,KTI-1 L<(,FC<N-L Finally, by Theorem 3.1, n is prime Q.E.D. Our next Lemma 3.3 aims at transforming the above Theorem 3.2 into a "trigonometric

1

1 primality criterion. Lemma 3.3. Let m,n be natural numbers with n > 1 and m = [\/n ]- Then 2 v- C

tj

k{l-C

k

) v "Intjk Z^ ^(n-1) I fk _ 2 Zv CO

b n l<j<m S ' S l<j<m Proof. The following hold true ,tjk

l

x „ . + 1

) . 7T& . TT* Trk(2tj + 1) (, J (1—C ) = 2 sin sin h sm — cos n n n n Also From (9) and (10) we obtain: sin iMMtil 2 V s * ^ ' = - V Z^ rfe(n-l) I rk _ 2 l<j<m S S 1 < j < m cos ,, , sin 1 < ] < m \<t,k<n- 1 Moreover E

sm — 3

+ . 27TÍ 7Ä.* irk a _ \ sin J— cot — sin — ^nn 1 <3<m n l<j<m l<(,fc< n-1 l<i,fc<n-l 1 <]<m Kt,k<n-1 (9) ^(«-i)

+

i*_2=-4sin

2—. (10) E

cus A • ("J cos-—-. (12)