# Representing Integers as Sums or Differences of Squares (UMAP)

### Author: Donald Malm

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. INTRODUCTION2. REPRESENTING AN INTEGER AS A DIFFERENCE OF SQUARES, I 3. CONGRUENCE AND RESIDUE ARITHMETIC4. REPRESENTING AN INTEGER AS A DIFFERENCE OF SQUARES, II5. THE AVERAGE NUMBER OF REPRESENTATIONS6. REPRESENTING AN INTEGER AS A SUM OF TWO SQUARES (A SELF-TEST) 7. REFERENCES8. ANSWERS TO EXERCISES9. ANSWERS FOR SECTION 6 (THE SELF-TEST) **

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