Irreducibility Criteria For Polynomials With Non-Negative Integer Coefficients, and the Prime Factorization of F(N) For F(X) In Z[X]

<p>We explore two specific connections between prime numbers and polynomials. Cohn's Criterion states that if d_nd_n-1\ldots d₀ is the base $10$ representation of a prime, then the polynomial \sum_j=0ⁿ 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 $2.52~\times~10³⁰\le c(10)\le 4.96~\times~10³¹.$ 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.</p>

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Campus Access Dissertation

Discipline thesis:degree_discipline

Mathematics

Year

2012

Author dc:creator

• Gross, Samuel S.

Contributors dc:contributor

• Michael Filaseta

• Mathematics
• Physical Sciences and Mathematics

Statement dc:rights

• © 2012, Samuel S. Gross

Repository record dc:identifier

https://scholarcommons.sc.edu/etd/1597

OAI identifier oai:identifier

oai:scholarcommons.sc.edu:etd-2598