  The traditional construction of primitive Pythagorean triples by the formulas
of two independent variables does not allow their ordering. The paper shows a
new view on the construction of primitive Pythagorean triples. A method for
constructing primitive Pythagorean triples, based on the use of gnomons equal
in area to the squares of the legs from a primitive Pythagorean triple, using
two dependent variables, is proposed. This enables the ability to build an
ordered table of primitive Pythagorean triples. Working with ordered data, it
is possible to approach in a new way the solution of problems that include
systems of Pythagorean triples, such as the construction of Eulerian
parallelepipeds, the construction of a Perfect Cuboid, or the problem of
coloring natural numbers in two colors so that no Pythagorean triple is
monochrome (one of the problems of Ramsey theory).
