## Date of Award

Spring 2020

## Document Type

Open Access Dissertation

## Department

Mathematics

## First Advisor

Michael Filaseta

## Abstract

This dissertation considers two different topics. In the first part of the dissertation, we show that a positive proportion of the primes have the property that if any one of their digits in base 10, including their infinitely many leading 0 digits, is replaced by a different digit, then the resulting number is composite. We show that the same result holds for bases b 2 {2, 3, · · · , 8, 9, 11, 31}.

In the second part of the dissertation, we show for an integer *b* ≥ 5 that if a polynomial ƒ( *x*) with non-negative coefficients satisfies the condition that ƒ( *b*) is prime, there are explicit bounds D_{4}(*b*) and D_{3}(*b*) so that if the degree of ƒ( *x*) is D_{4} then ƒ( *x*) is irreducible; and if the degree of ƒ( *x*) is D_{3} and ƒ( *x*) is reducible, then ƒ( *x*) is divisible by the shifted cyclotomic polynomial _{4}(*x-b*) Furthermore, in the case that b < 26, there are explicit bounds *D*_{6}(*b*) *D(b) *so that if the degree of ƒ( *x*) is ⋜ D₆ and ƒ( *x*) reducible, then ƒ( *x*) is divisible by either

## Rights

© 2020, Jeremiah T. Southwick

## Recommended Citation

Southwick, J. T.(2020). *Two Inquiries Related to the Digits of Prime Numbers.* (Doctoral dissertation). Retrieved from https://scholarcommons.sc.edu/etd/5879