## Theses and Dissertations

1-1-2012

#### Document Type

Campus Access Dissertation

#### Department

Mathematics

We explore two specific connections between prime numbers and polynomials. Cohn's Criterion states that if $d_nd_{n-1}\ldots d_0$ is the base $10$ representation of a prime, then the polynomial $\sum_{j=0}^n d_jx^j$ is irreducible. Let $f(x)$ be a polynomial with non-negative integer coefficients. We define $c(10)$ be the largest integer such that if $f(10)$ is prime and all the coefficients of $f(x)$ are $\le c(10)$, then $f(x)$ is irreducible. It is known that $$2.52~\times~10^{30}\le c(10)\le 4.96~\times~10^{31}.$$ We improve the lower bound above to $5.21~\times~10^{30}$. Furthermore, we classify all reducible polynomials $f(x)$ with non-negative integer coefficients such that $f(10)$ is prime and all the coefficients of $f(x)$ are $\le 4.96~\times~10^{31}$. Let $S$ be a finite set of rational primes. For a non-zero integer $n$, define $\left[ n\right] _{S}=\prod_{p\in S}\left\vert n\right\vert _{p}^{-1}$% , where $\left\vert n\right\vert _{p}$ is the usual $p$-adic norm of $n$. In 1984, Stewart applied Baker's theorem to prove non-trivial, computationally effective upper bounds for $[n(n+1)...(n+k)]_S$ for any integer $k>0$. Effective upper bounds have also been given by Bennett, Filaseta, and Trifonov for $[n(n+1)]_S$ and $[n^2+7]_S$, where $S=\{2,3\}$ and $S=\{2\}$, respectively. We extend Stewart's theorem to prove effective upper bounds for $[f(n)]_S$ for an arbitrary $f(x)$ in $\mathbb{Z}[x]$ having at least two distinct roots.