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November 2018 Polynomials in Base $x$ and the Prime-Irreducible Affinty
Fusun Akman
Missouri J. Math. Sci. 30(2): 197-217 (November 2018). DOI: 10.35834/mjms/1544151696

Abstract

Arthur Cohn's irreducibility criterion for polynomials with integer coefficients and its generalization connect primes to irreducibles, and integral bases to the variable $x$. As we follow this link, we find that these polynomials are ready to spill two of their secrets: (i) There exists a unique “base-$x$” representation of such polynomials that makes the ring $\mathbb{Z}[x]$ into an ordered domain; and (ii) There is a 1-1 correspondence between positive rational primes $p$ and certain infinite sets of irreducible polynomials $f(x)$ that attain the value $p$ at sufficiently large $x$, each generated in finitely many steps from the $p$th cyclotomic polynomial. The base-$x$ representation provides practical conversion methods among numeric bases (not to mention a polynomial factorization algorithm), while the prime-irreducible correspondence puts a new angle on the Bouniakowsky Conjecture, a generalization of Dirichlet's Theorem on Primes in Arithmetic Progressions.

Citation

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Fusun Akman. "Polynomials in Base $x$ and the Prime-Irreducible Affinty." Missouri J. Math. Sci. 30 (2) 197 - 217, November 2018. https://doi.org/10.35834/mjms/1544151696

Information

Published: November 2018
First available in Project Euclid: 7 December 2018

zbMATH: 07063855
MathSciNet: MR3884741
Digital Object Identifier: 10.35834/mjms/1544151696

Subjects:
Primary: 11A41
Secondary: 11A63, 11N32, 11Y05

Rights: Copyright © 2018 Central Missouri State University, Department of Mathematics and Computer Science

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Vol.30 • No. 2 • November 2018
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