Joshua Harrington

On the Iteration of a Function Related to Euler's φ-Function

Abstract A unit x in a commutative ring R with identity is called exceptional if 1 − x is also a unit in R. For any integer n ≥ 2, define φe (n) to be the number of exceptional units in the ring of integers modulo n. Following work of Shapiro, Mills, Catlin and Noe on iterations of Eulers φ-function, we develop analogous results on iterations of the function φe , when restricted to a particular subset of the positive integers.

Characterizing Finite Groups Using the Sum of the Orders of the Elements

We give characterizations of various infinite sets of finite groups under the assumption that and the subgroups of satisfy certain properties involving the sum of the orders of the elements of and . Additionally, we investigate the possible values for the sum of the orders of the elements of .

ON THE FACTORIZATION OF THE TRINOMIALS xn + cxn-1 + d

In this paper we investigate the factorization of trinomials of the form xn + cxn-1 + d ∈ ℤ[x]. We then use these results about trinomials to prove results about the factorization of polynomials of the form xn + c(xn-1 +⋯+ x + 1) ∈ ℤ[x].

Two questions concerning covering systems

Ever since Erdős introduced the concept of a covering system in 1950, many questions have arisen regarding the existence of certain types of covering systems. Two of the most famous questions regarding covering systems are the odd covering problem and the minimum modulus problem. In the current paper, we ask two questions that are related to these famous questions and provide results for each.

THE REDUCIBILITY OF CONSTANT-PERTURBED PRODUCTS OF CYCLOTOMIC POLYNOMIALS

In 1908, Schur raised the question of the irreducibility over ℚ of polynomials of the form f(x) = (x - a1)(x - a2)⋯(x - an) + 1, where the ai are distinct integers. Since then, many authors have addressed variations and generalizations of this question. In this article, we investigate the analogous question when replacing the linear polynomials with cyclotomic polynomials and allowing the constant perturbation of the product to be any integer d ∉ {-1, 0}. One interesting consequence of our investigations is that we are able to construct, for any positive integer N, an infinite set S of cyclotomic polynomials such that 1 plus the product of any k (not necessarily distinct) polynomials from S, where k ≢ 0(mod 2N+1), is reducible over ℚ.