#### Consortium for Mathematics and its Applications

Product ID: 99525
Supplementary Print

# 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.

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

9. ANSWERS FOR SECTION 6 (THE SELF-TEST) UMAP Module
32 pages

#### Mathematics Topics:

• Calculus ,
• Number Theory

#### Application Areas:

• Number Theory

#### Prerequisites:

Calculus (limits of sequences, the use of trigonometric substitutions to integrate)

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COMAP develops curriculum resources, professional development programs, and contest opportunities that are multidisciplinary, academically rigorous, and fun for educators and students. COMAP's educational philosophy is centered around mathematical modeling: using mathematical tools to explore real-world problems.

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