Values of coefficients of cyclotomic polynomials

Abstract

Let a(k, n) be the k-th coefficient of the n-th cyclotomic polynomials. In 1987, J. Suzuki proved that {a(k, n) | n, k ∈ N} = Z. In this paper, we improve this result and prove that for any prime p and any integer l ≥ 1, we have {a(k, p l n) | n, k ∈ N} = Z.

Remarks on a family of complex polynomials

“…In 1987, J. Suzuki [19] proves that in fact, any integer can be a coefficient of a cyclotomic polynomial of a certain degree. The result was further improved more recently by C.-G. Ji and W.-P. Li [13].…”

Values of coefficients of cyclotomic polynomials II

“…✷ Remark 1. If one specializes the above proof to the case m = p e , a proof a little easier than that given in part I [2] is obtained, since it does not involve a case distinction between m is odd and m is even as made in part I. This is a consequence of working modulo x 2p 1 , rather than modulo x 2p 1 +1 .…”

“…In 1987 Suzuki [4] proved that S(1) = Z. Recently the first two authors [2] proved that S(p e ) = Z with p e a prime power.…”

Inverse cyclotomic polynomials

“…It is not difficult, see Suzuki [22], to adapt his argument so as to show that every integer shows up as a coefficient, that is {a n (k): n 1, k 0} = Z. This was recently generalized by Ji and Li [11], who showed that if p e is a prime power, then {a p e n (k): n 1, k 0} = Z. This result on its turn was generalized by Ji, Li and Moree [12], who showed that, with m any positive integer, a mn (k): n 1, k 0 = Z and c mn (k): n 1, k 0 = Z.…”