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.
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.
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.
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 \(ax+by=c\) 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 \(ax+by=c\) where \(x,y\) are both positive, continuing our geometric intuition, in Section 3.3.
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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.