## Date of Award

Summer 2023

## Document Type

Open Access Dissertation

## Department

Mathematics

## First Advisor

Michael Filaseta

## Abstract

This dissertation considers three different sections of results. In the first part of the dissertation, a result on consecutive primes which are widely digitally delicate and Brier numbers is discussed. Making use of covering systems and a theorem of D. Shiu, M. Filaseta and J. Juillerat showed that for every positive integer *k*, there exist *k *consecutive widely digitally delicate primes. They also noted that for every positive integer* k*, there exist k consecutive primes which are Brier numbers. We show that for every positive integer *k*, there exist *k* consecutive primes that are both widely digitally delicate and Brier numbers. This is joint work with M. Filaseta and J. Juillerat.

In the second part of the dissertation, we prove an irreducibilty result for a class of polynomials. Consider the polynomial *F(x)* = *f(x)+Mg(x)* where *M* is a positive integer and *f(x)*, *g(x) ∈ Z*[*x*] such that gcd(*f(x)*, *g(x)*) = 1. A version of Hilbert’s Irreducibility Theorem in this setting implies that *F(x)* is irreducible for almost all M. In the case that deg *f* < deg *g*, recent results by M. Cavachi, M. Vajaitu, and A. Zaharescu [14] and by N.C. Bonciocat, Y. Bugeaud, M. Cipu, and M. Mignotte [9] have given definitive examples where irreducibility occurs by taking M to be a prime power bounded below by an explicit function depending on *f* and *g*. We provide a wider class of definitive examples by taking M with a large prime factor, and in particular our explicit examples include a set of M with positive asymptotic density in the integers. We then extend the result to bivariate polynomials in a manner similar to work by N.C. Bonciocat, Y. Bugeaud, M. Cipu, and M. Mignotte [9]. This is joint work with M. Filaseta.

* *

In the third part of the dissertation, we prove the irreducibility of nth order Euler polynomials of even degree n. For m an even positive integer and p a prime, we show that the generalized Euler polynomial

E^{(mp)}* *_{mp}* (x)* is in Eisenstein form with respect to p if and only if p does not divide m(2^{m} − 1)Bm. As a consequence, we deduce that at least 1/3 of the generalized Euler polynomials E^{(n)} _{n}*(x)* are in Eisenstein form with respect to a prime *p* dividing *n* and, hence, irreducible over Q. This is joint work with M. Filaseta.

## Recommended Citation

Luckner, T. D.(2023). *Widely Digitally Delicate Brier Primes and Irreducibility Results for Some Classes of Polynomials.* (Doctoral dissertation). Retrieved from https://scholarcommons.sc.edu/etd/7455