Date of Award

Spring 2021

Document Type

Open Access Dissertation

Department

Mathematics

First Advisor

Michael Filaseta

Abstract

This dissertation considers three different topics. In the first part of the dissertation, we show for an integer b > 2 that if a polynomial f(x) with non-negative integer coefficients is such that f(b) is prime, then there are explicit bounds M1(b), M2(b), and M3(b) such that if the coefficients of f(x) are each ≤ M1(b), then f(x) is irreducible; if the coefficients of f(x) are each ≤ M2(b) and f(x) is reducible, then it is divisible by the shifted cyclotomic polynomial Φ3(x−b) for 3 ≤ b ≤ 5, and divisible by Φ4(x − b) for b > 5; and if the coefficients of f(x) are each ≤ M3(b) and f(x) is reducible, then it is divisible by at least one of Φ3(x−b) and Φ4(x−b). Furthermore, if b > 69 and the coefficients of f(x) are each ≤ M4(b), then f(x) is either irreducible or divisible by at least one of Φ3(x − b), Φ4(x − b), and Φ6(x − b). In the second part of the dissertation, we show that there are only finitely many values of t such that the truncated binomial polynomial of degree 6,

q6,t(x) = X 6 j=0 t j ! x j .

has Galois group P GL2(5), a transitive subgroup of S6 isomorphic to S5. When the Galois group of the truncated binomial of degree 6 is not P GL2(5), it has been shown to be S6. Additionally, we show that the truncated binomial of degree 6 is irreducible for all values of t.

In the third part of the dissertation, we show that there are infinitely many composite numbers, N, with the property that inserting a digit between any two digits in base 10 of N, including between any two of the infinitely many leading zeros and to the right of N, always results in a composite number. We show that the same result holds for bases b ∈ {2, 3, · · · , 8, 9, 11, 31}.

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