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.
- 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.