A remark on flat ternary cyclotomic polynomials

2021 ◽  
Vol 56 (2) ◽  
pp. 241-261
Author(s):  
Bin Zhang ◽  
Keyword(s):  
Cyclotomic Polynomial ◽  
Cyclotomic Polynomials

Let \(\Phi_n(x)\) be the \(n\)-th cyclotomic polynomial. In this paper, for odd primes \(p\lt q \lt r\) with \(q\equiv \pm1\pmod p\) and \(8r\equiv \pm1\pmod {pq}\), we prove that the coefficients of \(\Phi_{pqr}(x)\) do not exceed \(1\) in modulus if and only if (i) \(p=3\), \(q\geq 19\) and \(q\equiv 1\pmod 3\) or (ii) \(p=7\), \(q\geq83\) and \(q\equiv -1\pmod 7\).

scholarly journals The order of magnitude of the mth coefficients of cyclotomic polynomials

1985 ◽  
Vol 27 ◽  
pp. 143-159 ◽  
Author(s):  
H. L. Montgomery ◽  
R. C. Vaughan
Keyword(s):  
Cyclotomic Polynomial ◽  
Cyclotomic Polynomials ◽  
Order Of Magnitude ◽  
Euler’S Function ◽  
Image Position

We define the nth cyclotomic polynomial Φn(z) by the equationand we writewhere ϕ is Euler's function.Erdös and Vaughan [3] have shown thatuniformly in n as m-→∞, whereand that for every large m

scholarly journals Modified Cyclotomic Polynomial and Its Irreducibility

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 343
Author(s):  
Ki-Suk Lee ◽  
Sung-Mo Yang ◽  
Soon-Mo Jung
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Cyclotomic Polynomial ◽  
Cyclotomic Polynomials ◽  
Irreducible Polynomials

Finding irreducible polynomials over Q (or over Z ) is not always easy. However, it is well-known that the mth cyclotomic polynomials are irreducible over Q . In this paper, we define the mth modified cyclotomic polynomials and we get more irreducible polynomials over Q systematically by using the modified cyclotomic polynomials. Since not all modified cyclotomic polynomials are irreducible, a criterion to decide the irreducibility of those polynomials is studied. Also, we count the number of irreducible mth modified cyclotomic polynomials when m = p α with p a prime number and α a positive integer.

A Generalization of the Cyclotomic Polynomial

1976 ◽  
Vol 19 (4) ◽  
pp. 461-466
Author(s):  
K. Nageswara Rao
Keyword(s):  
Fourier Series ◽  
Character Sums ◽  
Cyclotomic Polynomial ◽  
Cyclotomic Polynomials ◽  
Möbius Inversion

AbstractIn this paper, the cyclotomic polynomial is generalized and several of its properties based on the Môbius inversion are derived. It is deduced that a polynomial whose roots are the roots of a cyclotomic polynomial multiplied by those of another cyclotomic polynomial is the product of cyclotomic polynomials. Character sums and finite Fourier series are employed for the latter result.

Remarks on the flatness of ternary cyclotomic polynomials

2017 ◽  
Vol 13 (02) ◽  
pp. 529-547 ◽  
Author(s):  
Bin Zhang
Keyword(s):  
Positive Integer ◽  
Cyclotomic Polynomial ◽  
Cyclotomic Polynomials ◽  
Polynomial Formula

For a positive integer [Formula: see text], the [Formula: see text]th cyclotomic polynomial [Formula: see text] is called flat if its coefficients are [Formula: see text], [Formula: see text]. Let [Formula: see text] be odd primes with [Formula: see text] for some positive integer [Formula: see text]. In this paper, we classify all the flat cyclotomic polynomials [Formula: see text] when [Formula: see text], [Formula: see text] and [Formula: see text].

The maximal coefficient of ternary cyclotomic polynomials with one free prime

2014 ◽  
Vol 10 (04) ◽  
pp. 1067-1080 ◽  
Author(s):  
Dominik Duda
Keyword(s):  
Cyclotomic Polynomial ◽  
Cyclotomic Polynomials ◽  
Absolute Value ◽  
Main Conjecture ◽  
Polynomial Family

A cyclotomic polynomial Φn(x) is said to be ternary if n = pqr, with p, q and r distinct odd primes. Let M(p, q) be the maximum (in absolute value) coefficient appearing in the polynomial family Φpqr(x) with p < q < r, p and q fixed. Here a stronger version of the main conjecture of Gallot, Moree and Wilms regarding M(p, q) is established. Furthermore it is shown that there is an algorithm to compute M(p): = max {M(p, q): q > p}. Our methods are the most geometric used so far in the study of ternary cyclotomic polynomials.

A Note on Golomb's “Cyclotomic Polynomials and Factorization Theorems”

1981 ◽  
Vol 88 (10) ◽  
pp. 753-753
Author(s):  
Katherine E. McLain ◽  
Hugh M. Edgar
Keyword(s):  
Cyclotomic Polynomials

scholarly journals On modular inverses of cyclotomic polynomials and the magnitude of their coefficients

2012 ◽  
Vol 15 ◽  
pp. 44-58 ◽  
Author(s):  
Clément Dunand
Keyword(s):  
Upper Bounds ◽  
Cyclotomic Polynomials ◽  
Lower And Upper Bounds

AbstractLet p and r be two primes, and let n and m be two distinct divisors of pr. Consider Φn and Φm, the nth and mth cyclotomic polynomials. In this paper, we present lower and upper bounds for the coefficients of the inverse of Φn modulo Φm and discuss an application to torus-based cryptography.

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