Composite Numbers in an Arithmetic Progression

TL;DR

This paper explores a family of number theory problems based on Dirichlet's Theorem, designed to accommodate diverse student backgrounds and promote research-level exploration.

Contribution

It introduces a set of problems illustrating various approaches to Dirichlet's Theorem, fostering inclusive and research-oriented learning in number theory.

Findings

Problems highlight different solution strategies

Encourage student extension and exploration

Bridge classroom learning with research-level mathematics

Abstract

One challenge (or opportunity!) that many instructors face is how varied the backgrounds, abilities, and interests of students are. In order to simultaneously instill confidence in those with weaker preparations and still challenge those able to go faster, an instructor must be prepared to give problems of different difficulty levels. Using Dirichlet's Theorem as a case study, we create and discuss a family of problems in number theory that highlight the relative strengths and weaknesses of different ways to approach a question and show how to invite students to extend the problems and explore research-level mathematics.