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Explorations In Elementary And Analytic Number Theory, Scott Michael Dunn

Theses and Dissertations

In this dissertation, we investigate two distinct questions in number theory. Each question is dedicated its own chapter.

First, we consider arithmetic progressions in the polygonal numbers with a fixed number of sides. We will show that four-term arithmetic progressions cannot exist. We then describe explicitly how to find all three-term arithmetic progressions. Additionally we show that there are infinitely many three-term arithmetic progressions starting with an arbitrary polygonal number.

Second, we will show certain irreducibility criteria for polynomials. Let f(x) be a polynomial with non-negative integer coefficients such that f(b) is prime for some integer 2 ≤ b ≤ …