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An Analysis Of Polynomials That Commute Under Composition, Samuel J. Williams
An Analysis Of Polynomials That Commute Under Composition, Samuel J. Williams
Honors Theses
It is well known that polynomials commute under addition and multiplication. It turns out that certain polynomials also commute under composition. In this paper, we examine polynomials with coefficients in the field of complex numbers that commute under composition (also referred to as “commuting polynomials”). We begin this examination by defining what it means for polynomials to commute under composition. We then introduce sequences of commuting polynomials and observe how the polynomials in these sequences (later defined as chains) along with other commuting polynomials relate to a concept called similarity. These observations allow us to better understand the qualities and …
Selected Research In Covering Systems Of The Integers And The Factorization Of Polynomials, Joshua Harrington
Selected Research In Covering Systems Of The Integers And The Factorization Of Polynomials, Joshua Harrington
Theses and Dissertations
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 …