Additional Thoughts on the Pythagorean Triple Generator

A prior post list a formula for producing PythagoreanTriples. A Pythagorean Triple is a set of three integers (a, b, c) that will form a right angle triangle. The formula is:

Let n and m be any positive integers where n>m. The Pythagorean Triple generated is formed by the numbers:
2nm
n² - m²
n² +m²

This prior post did not state way this formula works. We can show why in just a few steps. The two requirements for a Pythagorean Triple (a, b, c) are that a, b, and c are integers and that a² + b² = c².

For the first requirement, we need to show that 2nm,  n² - m² , and  n² +m² are all integers. It is given that n and m are integers, Since both n and m are integers, a product of two integers is an integer and 2 times an integer is an integer, so 2nm is an integer. Likewise, for n² - m²
and n² +m², we are only dealing with squares of integers, their sums and their differences. All these operations will produce only integers, therefore n² - m² and n² +m² are also integers.

For the second requirement, we need to show that (2nm)² + (n² - m²)² = (n² +m²)²

Expanding the left-hand side,

 (2nm)² + (n² - m²)² = 4n²m² + n⁴ – 2n²m² + m⁴

Simplifying and reordering,       

                              = n⁴ + 2n²m² + m⁴

Compare this result to squaring the original right-hand side, we see the Pythagorean Triple generator formula will produce triples that meet the requirements of being proper Pythagorean Triples.

(n² +m²)² = n⁴ + 2n²m² + m⁴

^{Update 9/24/2022: A spreadsheet has been created to produce these triples.}