Selected Research In Covering Systems of the Integers and the Factorization of Polynomials

Publication Year:
2013
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Repository URL:
https://scholarcommons.sc.edu/etd/2434
Author(s):
Harrington, Joshua
Tags:
Mathematics; Systems of the Integers; Factorization of Polynomials; integers; Polynomials; Physical Sciences and Mathematics
interview description
In 1960, Sierpi\'{n}ski proved that there exist infinitely many odd positive integers $k$ such that $k\cdot 2^n+1$ is composite for all positive integers $n$. Such integers are known as Sierpi\'{n}ski numbers. Letting $f(x)=ax^r+bx+c\in\mathbb{Z}[x]$, Chapter 2 of this document explores the existence of integers $k$ such that $f(k)2^n+d$ is composite for all positive integers $n$. Chapter 3 then looks into a polynomial variation of a similar question. In particular, Chapter~\ref{CH:FH} addresses the question, for what integers $d$ does there exist a polynomial $f(x)\in\mathbb{Z}[x]$ with $f(1)\neq -d$ such that $f(x)x^n+d$ is reducible for all positive integers $n$. The last two chapters of the document then explore the reducibility and factorization of polynomials taking on a prescribed form. Specifically, Chapter 4 addresses the reducibility and factorization of polynomials of the form $x^n+cx^{n-1}+d\in\mathbb{Z}[x]$, while Chapter 5 addresses the reducibility and factorization of polynomials of the more general form $f(x)x^n+g(x)\in\mathbb{Z}[x]$.