The Midterm Coefficient of the Cyclotomic Polynomial F pq (x)

@article{Beiter1964TheMC,
  title={The Midterm Coefficient of the Cyclotomic Polynomial F pq (x)},
  author={Marion Beiter},
  journal={American Mathematical Monthly},
  year={1964},
  volume={71},
  pages={769}
}
  • M. Beiter
  • Published 1 August 1964
  • Mathematics
  • American Mathematical Monthly
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On the Coefficients of Cyclotomic Polynomials
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Cyclotomic polynomials play an important role in several areas of mathematics and their study has a very long history, which goes back at least to Gauss (1801). In particular, the properties of their…
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Abstract. The scaled inverse of a nonzero element a(x) ∈ Z[x]/f(x), where f(x) is an irreducible polynomial over Z, is the element b(x) ∈ Z[x]/f(x) such that a(x)b(x) = c (mod f(x)) for the smallest…
On Arithmetic Progressions of Powers in Cyclotomic Polynomials
  • H. Chu
  • Computer Science, Mathematics
    Am. Math. Mon.
  • 2021
TLDR
The necessary conditions for when powers corresponding to positive/negative coefficients of are in arithmetic progression are determined and the result when n = pq is generalized to the so-called inclusion-exclusion polynomials first introduced by Bachman.
On the Scaled Inverse of $(x^i-x^j)$ modulo Cyclotomic Polynomial of the form $\Phi_{p^s}(x)$ or $\Phi_{p^s q^t}(x)$
Abstract. The scaled inverse of a nonzero element a(x) ∈ Z[x]/f(x), where f(x) is an irreducible polynomial over Z, is the element b(x) ∈ Z[x]/f(x) such that a(x)b(x) = c (mod f(x)) for the smallest…
Representation of $\frac{1}{2}(F_n-1)(F_{n+1}-1)$ and $\frac{1}{2}(F_n-1)(F_{n+2}-1)$
Let $a, b\in \mathbb{N}$ be relatively prime. We consider $(a-1)(b-1)/2$, which arises in the study of the $pq$-th cyclotomic polynomial, where $p,q$ are distinct primes. We prove two possible…
Explicit expression for a family of ternary cyclotomic polynomials
In this paper, we give an explicit expression for a certain family of ternary cyclotomic polynomials: specifically $\Phi_{p_{1}p_{2}p_{3}}$, where $p_{1}<p_{2}<p_{3}$ are odd primes such that $p_{2}…
On the Middle Coefficient of a
Numerical Semigroups, Cyclotomic Polynomials, and Bernoulli Numbers
  • P. Moree
  • Mathematics, Computer Science
    Am. Math. Mon.
  • 2014
TLDR
The intent of this paper is to better unify the various results within the cyclotomic polynomial and numerical semigroup communities.
Constants of cyclotomic derivations
Let k[X]=k[x0,…,xn−1] and k[Y]=k[y0,…,yn−1] be the polynomial rings in n⩾3 variables over a field k of characteristic zero containing the n-th roots of unity. Let d be the cyclotomic derivation of…
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References

Introduction to Number Theory
A specific feature of this text on number theory is the rather extensive treatment of Diophantine equations of second or higher degree. A large number of non-routine problems are given. The book is…