  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.
