## Theses and Dissertations

1-1-2013

#### Document Type

Open Access Thesis

Mathematics

Michael Filaseta

Ognian Trifonov

#### Abstract

One may easily show that there exist $O( \log n)$-colorings of $\{1,2, \ldots, n\}$ such that no Pythagorean triple with elements $\le n$ is monochromatic. In Chapter~\ref{CH:triples}, we investigate two analogous ideas. First, we find an asymptotic bound for the number of colors required to color $\{1,2,\ldots ,n\}$ so that every Pythagorean triple with elements $\le n$ is $3$-colored. Afterwards, we examine the case where we allow a vanishing proportion of Pythagorean triples with elements $\le n$ to fail to have this property.

Unrelated, in 1908, Schur raised the question of the irreducibility over $\Q$ of polynomials of the form $f(x)=(x-a_1)(x-a_2)\cdots (x-a_n)+1$, where the $a_i$ are distinct integers. Since then, many authors have addressed variations and generalizations of this question. In Chapter~\ref{CH:polynomials}, we investigate the analogous question when replacing the linear polynomials with cyclotomic polynomials and allowing the constant perturbation of the product to be any integer $d\not \in \{-1,0\}$.

COinS