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Composite Numbers in an Arithmetic Progression

Hung Viet Chu, Steven J. Miller, Joshua M. Siktar

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.

Composite Numbers in an Arithmetic Progression

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.

Paper Structure

This paper contains 6 sections, 3 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Solution
  3. Asking more refined questions
  4. Relating our problem to advanced results
  5. Conclusion
  6. Proof of Proposition \ref{['Prop: ZeroDense']}

Key Result

Theorem 1.1

If $a$ and $b$ are relatively primeSometimes one assumes that our numbers are positive; we leave it to the reader to show that such an assumption is harmless. then there are infinitely many primes congruent to $b$ mod $a$. In other words, the sequence $an + b$ is prime for infinitely many $n$.

Theorems & Definitions (7)

  • Theorem 1.1: Dirichlet
  • Proposition 1.2: Telhcirid
  • Proposition 2.1: Zero density of primes
  • proof : (First) proof of Proposition \ref{['Prop: Telhcirid']}
  • proof : (Second) proof of Proposition \ref{['Prop: Telhcirid']}
  • proof : Third proof of Proposition \ref{['Prop: Telhcirid']}
  • proof : Proof of Proposition \ref{['Prop: ZeroDense']}