ON A CLASS OF TERNARY CYCLOTOMIC POLYNOMIALS

Abstract

Abstract. A cyclotomic polynomial Φ n (x) is said to be ternary if n=pqr for three distinct odd primes p < q < r. Let A(n) be the largestabsolute value of the coefficients of Φ n (x). If A(n) = 1 we say that Φ n (x)is flat. In this paper, we classify all flat ternary cyclotomic polynomialsΦ pqr (x) in the case q≡±1 (mod p) and 4r≡±1 (mod pq). 1. IntroductionLetΦ n (x) =Y n k=1 (k,n)=1 (x −e 2πikn ) = φ X ( )j=0 a(n,j)x j be the n-th cyclotomic polynomial, where φ is the Euler totient function. Itcan be shown that a(n,j) ∈ Z. LetA(n) = max{|a(n,j)| | 0 ≤ j ≤ φ(n)}denote the largest absolute value of the coefficients of Φ n (x). If A(n) = 1 wesay that Φ n (x) is flat. It turns out that for the purpose of determining A(n),it suffices to consider squarefree and odd integers n. Clearly, if n has at mosttwo distinct odd prime factors, then A(n) = 1.However, the coefficients of ternary cyclotomic polynomials Φ pqr (x), wherep < q < r areodd primes, become much morecomplicated, such asa(3·5·7,7) =−2 and a(5·7·11,119) = −3. The investigation about the coefficients of ternarycyclotomic polynomials have a long history and there are many references onthis subject, see, for instance, [1-17, 19, 20, 21, 23, 24]. One interesting openproblem involving this topic is to give a complete characterization of all flatternary cyclotomic polynomials, but this appears very difficult. Throughout