NTIC From Linear Equations to Geometry

Chapter 3 From Linear Equations to Geometry

So far, we have mostly investigated topics that will seem familiar even to the high school student; for instance, the gcd shows up in adding fractions with unequal denominators.

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What makes number theory so interesting is that even a slight change in the questions we ask, or the way in which we approach them, can yield completely unexpected insights.

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In this section, we will begin this process by going from the simple questions we started with into more subtle ones, largely motivated by a surprising connection with geometry.

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Summary: From Linear Equations to Geometry

This chapter contains a lot of interesting results about equations involving integers, including a number of geometric interpretations.

In Solutions of Linear Diophantine Equations we solve all equations of the form $a x + b y = c$ in integers. There are several cases, the most important being where $c = gcd (a, b)$ in Subsection 3.1.3.
The next section reinterprets these results gometrically, using the integer lattice.
Then we try to ask for solutions to $a x + b y = c$ where $x, y$ are both positive, continuing our geometric intuition, in Section 3.3.
Moving to equations with quadratic terms, we introduce the notion of Pythagorean triples.
- We prove the Characterization of primitive Pythagorean triples.
- We also examine the possible areas of integer-sided right triangles in Subsection 3.4.3, including the historically very important question of whether such areas can themselves be a perfect square.
In the last main section, we start examining further interesting questions such as the Bachet equation and Catalan's Conjecture.

Finally, after a good selection of Exercises, we have proofs of two facts about perfect squares using just the machinery available to us at this time.