NON-BEITER TERNARY CYCLOTOMIC POLYNOMIALS WITH AN OPTIMALLY LARGE SET OF COEFFICIENTS

作者：Moree Pieter^*; Rosu Eugenia

来源：International Journal of Number Theory, 2012, 8(8): 1883-1902.

DOI：10.1142/S1793042112501072

• Coefficient
• optimally large
• ternary cyclotomic polynomial
• Beiter conjecture
• Kaplan's lemma

Let l >= 1 be an arbitrary odd integer and p, q and r be primes. We show that there exist infinitely many ternary cyclotomic polynomials phi(pqr)(x) with l(2) + 3l + 5 <= p < q < r such that the set of coefficients of each of them consists of the p integers in the interval [ - (p - l - 2)/2, (p + l + 2)/2]. It is known that no larger coefficient range is possible. The Beiter conjecture states that the cyclotomic coefficients a(pqr)(k) of phi(pqr) satisfy vertical bar a(pqr)(k)vertical bar <= (p + 1)/2 and thus the above family contradicts the Beiter conjecture. The two already known families of ternary cyclotomic polynomials with an optimally large set of coefficients (found by G. Bachman) satisfy the Beiter conjecture.

• 出版日期2012-12