Geometric and algebraic interpretation of primitive Pythagorean triples parameters

TL;DR

This paper provides a geometric and algebraic interpretation of the parameters m and n in primitive Pythagorean triples, using the concept of gnomons and figurate numbers to deepen understanding of their construction.

Contribution

It introduces a novel geometric approach to interpret the parameters m and n in primitive Pythagorean triples through gnomons and figurate number constructions.

Findings

Gnomon U corresponds to the area of y^2 in the triples.

Construction of squares using gnomons reveals relationships between parameters.

Formulas for m and n derived from partitioning the generating square.

Abstract

The paper found a geometric and algebraic interpretation of the parameters m and n from the formulas for obtaining primitive Pythagorean triples, which are solutions of the equation ${x^2+y^2=z^2}$, namely: ${x=m^2-n^2}$, ${y=2mn}$, ${z=m^2+n^2}$. The study was based on the process of building figurate numbers using gnomons. The paper discusses the process of building squares. The addition of the gnomon U to the original square leads to a larger square: ${x^2+U=z^2}$. The first stage of the investigation was the construction of the gnomon U, which is equal to the area of a square ${y^2}$. The construction is based on a generating square with a side equal to an even number. The area of the generating square is represented as the sum of the areas of two equal rectangles in all possible ways. At the same time, using the generating square, the gnomon U and the sides of all squares are also constructed: x, y, z. The second stage of the investigation was to obtain a formula for the parameters m and n through the partition elements t and l of the side of the generating square.