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

Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004) 3-10 PRIME NUMBERS AND CYCLOTOMY Panayiotis G. Tsangaris (Athens, Greece) Abstract. First, an explicite expression for (

1 —

C* )

_

I

, where £=exp(27ri'//i

), is given, in the form of a polynomial in f, with rational coefficients. Then a new primality criterion is obtained, which involves the greatest integer function. Further, using a result due to Yu.I. Volosin [1U], we transform this criterion into a series of criteria involving rational expressions of C, [one of these criteria involves the numbers (1 — C*)1* l<fc<n —l]. Finally, these criteria are refined to a trigonometric primality criterion, that involves only sums of cosines. AMS Classification Number: 1 I A5 I

, 111118 Introduction Denote by F

n

(x

) the n-th cyclotomic polynomial, while 0 will denote Euler's function and ( = exp(27n/ra). Given two polynomials f(v), ä(

v) m variable v, denote by fí

r(f(v), g(v)) tlieir resultant. In Section 1 we express (1 — C

A )1? explicitly, in the form of a polynomial in <,", by employing a series of new properties of the cyclotomic polynomial (Theorems 1.1 and 1.2). In Section 2 a new primality criterion is obtained. Our primality criterion (Theorem 2.1) extends a previous result of author [7] which improves upon classical result of Hacks [5]. In Section 3 the result of (Section 2) is given in "cyclotomic" form by using roots of unity and trigonometric functions. The key result for such a ""cyclotomic" modification is a Theorem of Yu. i. Volosin [10] expressing [a/n] by means of a primitive root of 1 of order n. Specifically, our Theorem 3.1 is a first primality criterion for n formulated in terms of (,* and involving (1 — C^)1) \ < k < n — 1. To calculate the inverse of (1 — (

k

) (Corollary 1.4), we thus obtain a second "cyclotomic" primality criterion (Theorem 3.2). The "trigonometric elaboration" of this result leads to our final Theorem 3.4, which is a "trigonometric" primality criterion. 1. Expressing (1 — C

A*)

_1 as a polynomial in (," Theorem 1.1. Let n, s be natural numbers and let d— (n,s). Then