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 ≤ 20. A. Cohn\u27s criteria states that if b=10 and each coefficient is ≤ 9, then f(x) is irreducible. In 1988, M. Filaseta showed that the bound 9 can be replaced by 1030. We will look at work that was done to further increase this bound and then generalize this for an arbitrary base b. Along the way, we will also establish additional irreducibility criteria
