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Values of coefficients of cyclotomic polynomials II
, 2007
"... Let a(n,k) be the kth coefficient of the nth cyclotomic polynomial. In part I it was proved that {a(mn,k)  n ≥ 1, k ≥ 0} = Z, in case m is a prime power. In this paper we show that the result also holds true in case m is an arbitrary positive integer. ..."
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Let a(n,k) be the kth coefficient of the nth cyclotomic polynomial. In part I it was proved that {a(mn,k)  n ≥ 1, k ≥ 0} = Z, in case m is a prime power. In this paper we show that the result also holds true in case m is an arbitrary positive integer.
Reciprocal cyclotomic polynomials
, 2007
"... Let Ψn(x) be the monic polynomial having precisely all nonprimitive nth roots of unity as its simple zeros. One has Ψn(x) = (x n − 1)/Φn(x), with Φn(x) the nth cyclotomic polynomial. The coefficients of Ψn(x) are integers that like the coefficients of Φn(x) tend to be surprisingly small in absolut ..."
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Cited by 4 (2 self)
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Let Ψn(x) be the monic polynomial having precisely all nonprimitive nth roots of unity as its simple zeros. One has Ψn(x) = (x n − 1)/Φn(x), with Φn(x) the nth cyclotomic polynomial. The coefficients of Ψn(x) are integers that like the coefficients of Φn(x) tend to be surprisingly small
COEFFICIENTS OF THE CYCLOTOMIC POLYNOMIAL F3qr (*)
"... Let Fm be the 777th cyclotomic polynomial. Bang [1] has shown that for m = pqv> a product of three odd primes with p < q < r, tile coefficients of Fm(x) do not exceed p 1 in absolute value. The smallest such m is 105 and the coefficient of x 7 in F105 is2. It might be assumed that coeffic ..."
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Let Fm be the 777th cyclotomic polynomial. Bang [1] has shown that for m = pqv> a product of three odd primes with p < q < r, tile coefficients of Fm(x) do not exceed p 1 in absolute value. The smallest such m is 105 and the coefficient of x 7 in F105 is2. It might be assumed
On Cyclotomic Polynomials with ±1 Coefficients
"... We characterize all cyclotomic polynomials of even degree with coefficients restricted to the set f+1; \Gamma1g. In this context a cyclotomic polynomial is any monic polynomial with integer coefficients and all roots of modulus 1. Inter alia we characterize all cyclotomic polynomials with odd coef ..."
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We characterize all cyclotomic polynomials of even degree with coefficients restricted to the set f+1; \Gamma1g. In this context a cyclotomic polynomial is any monic polynomial with integer coefficients and all roots of modulus 1. Inter alia we characterize all cyclotomic polynomials with odd
On the Coefficients of Cyclotomic Polynomials
"... Cyclotomy is the process of dividing a circle into equal parts, which is precisely the effect obtained by plotting the nth roots of unity in the complex plane. For integers n ≥ 1, we know that Xn−1 = ∏n−1m=0(X−e 2piimn) over C. The nth cyclotomic polynomial can be defined as Φn(X) = n∏ m=1,(m,n)=1 ..."
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Cited by 2 (0 self)
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Cyclotomy is the process of dividing a circle into equal parts, which is precisely the effect obtained by plotting the nth roots of unity in the complex plane. For integers n ≥ 1, we know that Xn−1 = ∏n−1m=0(X−e 2piimn) over C. The nth cyclotomic polynomial can be defined as Φn(X) = n∏ m=1,(m
Coefficients of cyclotomic polynomials
, 2009
"... Let a(n, k) be the kth coefficient of the nth cyclotomic polynomial. Recently, Ji, Li and Moree [12] proved that for any integer m ≥ 1, {a(mn, k)n, k ∈ N} = Z. In this paper, we improve this result and prove that for any integers s> t ≥ 0, {a(ns + t, k)n, k ∈ N} = Z. ..."
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Let a(n, k) be the kth coefficient of the nth cyclotomic polynomial. Recently, Ji, Li and Moree [12] proved that for any integer m ≥ 1, {a(mn, k)n, k ∈ N} = Z. In this paper, we improve this result and prove that for any integers s> t ≥ 0, {a(ns + t, k)n, k ∈ N} = Z.
ZEROS OF POLYNOMIALS WITH CYCLOTOMIC COEFFICIENTS
"... This paper examines properties of the zeros of polynomials with restricted coefficients. In particular we study the case when the coefficients are restricted to the roots of unity and possibly zero. The methods used in this paper are adaptations of methods used by Odlyzko and Poonen in “Zeros of Po ..."
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This paper examines properties of the zeros of polynomials with restricted coefficients. In particular we study the case when the coefficients are restricted to the roots of unity and possibly zero. The methods used in this paper are adaptations of methods used by Odlyzko and Poonen in “Zeros
On Prime Values of Cyclotomic Polynomials
"... We present several approaches on finding necessary and sufficient conditions on n so that Φk(xn) is irreducible where Φk is the kth cyclotomic polynomial. ..."
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Cited by 3 (1 self)
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We present several approaches on finding necessary and sufficient conditions on n so that Φk(xn) is irreducible where Φk is the kth cyclotomic polynomial.
ON VALUES OF CYCLOTOMIC POLYNOMIALS. VII
, 2004
"... In this paper, we present three results on cyclotomic polynomials. First, we present results about factorization of cyclotomic polynomials over arbitrary fields K. It is well known in cases such that a field K is the rational number field Q or a finite field F q (see [3, 4]). Using irreducibility of ..."
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In this paper, we present three results on cyclotomic polynomials. First, we present results about factorization of cyclotomic polynomials over arbitrary fields K. It is well known in cases such that a field K is the rational number field Q or a finite field F q (see [3, 4]). Using irreducibility
Results 1  10
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1,225