Representing Integers as Sums or Differences of Squares (UMAP)

In this module the problem of presenting an integer as a sum or as a difference of two squares is studied. Key questions considered are: which integers can be so represented? and what is the behavior of the average number of ways an integer can be represented? Students 1) practice and develop problem-solving skills; 2) become acquainted with the idea of congruence and its uses; 3) use the computer to gather facts about a problem; and 4) use geometry and calculus to help solve problems of representing integers.

Table of Contents:

1. INTRODUCTION

2. REPRESENTING AN INTEGER AS A DIFFERENCE OF SQUARES, I

3. CONGRUENCE AND RESIDUE ARITHMETIC

4. REPRESENTING AN INTEGER AS A DIFFERENCE OF SQUARES, II

5. THE AVERAGE NUMBER OF REPRESENTATIONS

6. REPRESENTING AN INTEGER AS A SUM OF TWO SQUARES (A SELF-TEST)

7. REFERENCES

8. ANSWERS TO EXERCISES

9. ANSWERS FOR SECTION 6 (THE SELF-TEST)