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 D4(b) and D3(b) so that if the degree of ƒ( x) is D4 then ƒ( x) is irreducible; and if the degree of ƒ( x) is D3 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 D6(b) D(b) so that if the degree of ƒ( x) is ⋜ D₆ and ƒ( x) reducible, then ƒ( x) is divisible by either

Included in

Mathematics Commons

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