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Natalia Aleshkevich

Publications and source records attributed to Natalia Aleshkevich.

4 recordsLinked to original sources
arXiv2022

Construction of primitive Pythagorean triples and Pythagorean triples with a common multiplier using gnomons

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).

math.GM↗
arXiv2021

Constructive representation of primitive Pythagorean triples

The paper presents a systematic construction of primitive Pythagorean triples. The order of enumeration on the set of primitive Pythagorean triples is defined. The order is based on the representation of a primitive Pythagorean triple by dividing the side of the generating square into two groups of factors using gnomons. In paper is shown the inverse mapping of the elements of a primitive Pythagorean triple to the parameters of the partition of the side of the generating square, the representation of primitive Pythagorean triples as the sum of two connected gnomons, a one-to-one correspondence of the connected gnomons of the primitive Pythagorean triple with the corresponding arithmetic progressions. Also the property of such arithmetic progressions as connectedness with each other, namely, partial overlap, is found. The motion (configuration change) of the connected gnomons is shown during the transition of a primitive Pythagorean triple to a general Pythagorean triple, all the terms of which have one common coefficient.

math.NT↗
arXiv2019

Geometric and algebraic interpretation of primitive Pythagorean triples parameters

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.

math.NT↗