Mathematical Proofs

Here are some things that I wanted to try to prove.

Theorem: There exist an infinite number of pythagorean triples
Definition: A pythagorean triple is a set of three natural numbers $a$ , $b$ , and $c$ that satisfy the equation $a^2 + b^2 = c^2$

To show that there are an infinite number of triples, we only need to show that there are at least an infinite number of triples in a certain form. Let $a = n^2$ and $c = (n+1)^2$ where $n$ is a natural number. From $a^2 + b^2 = c^2$ , $c^2 - a^2 = b^2$ , and $(n+1)^2 - n^2 = b^2$ . Expanding the left side of the equation results in $2n+1 = b^2$ . Since $2n+1$ is an odd number for all $n$ , and $b^2$ is odd for all odd $b$ , and because the intersection of the sest of all $2n+1$ and of all $b^2$ is infinitely large, there are an infinite number of pythagorean triples in the form $n^2 + b^2 = (n+1)^2$ , and therefore an infinite number of pythagorean triples.