## Theses and Dissertations

Spring 2021

#### Document Type

Open Access Dissertation

Mathematics

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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