The theory behind Pythagorean triples

In this article, we’re going to go through the theory of Pythagorean triples. When exactly can we have a Pythagorean triple? Can we generate these? Can we find Pythagorean triples with specific values of our choice? These are all questions that can be answered, and we are going to do our best to do so. Along the way, the reader may see (**) which marks parts that the reader should attempt themselves, to fully immerse themselves into the mathematics. This is where we begin.

First, let us define what we mean by a Pythagorean triple. A Pythagorean triple is a triple such that x,y,z are integers and we have x² + y² = z² . We call this triple primitive if it gcd(x,y,z) = 1. The most famous example of this is 3,4,5 since 3² + 4² = 5² . We can scale triples such as this and we still get what we want. For example, consider multiplying the whole equation by 2². Then we get 6² + 8² = 10² . That is, we can generate new Pythagorean triples from old ones.

We now prove that if we have a primitive Pythagorean triple, then exactly one of x or y is odd, and the other is even (**)

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Suppose that they are both odd. Then we have (2m+1)² + (2n+1)² = z² for some integer m,n. Expanding out we yield 4m² + 4m + 4n² + 4n + 2 = z². Clearly z² must be even and so z is even, meaning that we can write z = 2k. Thus we have 4m² + 4m + 4n² + 4n + 2 = 4k² . However, this implies that 2 is divisible by 4, which it is not!
What if we instead assume…