PASCAL TRIANGLE AND PYTHAGOREAN TRIPLES

JP Journal of Algebra, Number Theory and Applications, 2017

The main aim of this paper is to present an analytic result which characterizes the Pythagorean triples via a cathetus. This way has the convenience to find easily all Pythagorean triples , , , N ∈ z y x where x is a predetermined integer, which means finding all right triangles whose sides have integer measures and one cathetus is predetermined.

2015

The standard formula for the production of all possible primitive Pythagorean triples [3, pp.237] is well known. There is an alternative, however, (2 x qr + 2 2x−1 q 2) 2 + (2 x qr + r 2) 2 = (2 x qr + 2 2x−1 q 2 + r 2) 2 , (1) x a positive integer, q and r relatively prime positive odd integers, or q = 1 and r = 1, q = 1 and r = 1, or q = r = 1. After developing equation (1), I discovered that it had been previously worked out by Bottari [2, pp.169]. Below I present a proof independent of his that equation (1) generates all possible primitive Pythagorean triples. First I present and demonstrate the validity of an alternative to the standard binomial expansion formula [3, pp.9]. Then, in the proof proper, I use the alternative binomial formula to show that equation (1) generates all possible primitive Pythagorean triples.

arXiv (Cornell University), 2015

In 1987, Orrin Frink introduced the concept of almost Pythagorean triples. He defined them as an ordered triple (x, y, z) that satisfies the Diophantine equation x 2 + y 2 = z 2 + 1 where x, y and z are positive integers. In his paper, he determined all almost Pythagorean triples by solving the given Diophantine equation. However, the result does not explicitly and readily give a particular almost Pythagorean triple. In this note, using basic algebraic operations and Frink's result we give an explicit formula that readily gives a particular almost Pythagorean triple. We also give a certain integer sequence as a result of the generated formula.

This paper revisits the topic of Pythagorean triples with a different perspective. While several methods have been explored to generate Pythagorean triples, none of them is complete in terms of generating all the triples without repetitions. Indeed, many existing methods concentrate on generating primitive triples but do not cater to non-primitives. By contrast, the approach presented in this paper to parameterise the Pythagorean triples generates all of the triples in a unique way, i.e., without repetitions. We also explore the relation of this new parameterisation with the Pythagorean family of odd triples and with the Platonic family of even triples.

We obtain non-trivial values for the sides of the Pythagorean triangle such that () () (). A few interesting relations between the sides of the Pythagorean triangle are presented.

Parabola Volume 59, Issue 2, 2023

Introducing a novel method for generating multiple families of Pythagorean triples using sequences of fractions derived from periodic continued fractions.

arXiv (Cornell University), 2008

To find out the Special Pythagorean Triangles, where their perimeters are Triangular numbers and Pentagonal numbers both is the main objective of this paper. A few interesting results are observed.

International Journal of Applied Research

In this paper we have introduced a new approach tantamounting the regular one. It helps derive some additional algebraic properties. The novel concept of finding dual triplets of a given triplet and its generalization is an important feature of this note. Some new observations on ratios of corresponding sides of consecutive odd Pythagorean triplets and numerical properties related to 'Jha sequence' and 'NSW sequence' are explicitly demonstrated and derived in three annexure. Keywords (1) Primitive Pythagorean triplets (2) Dual triplets of a given triplet (3) Ratio of corresponding sides (4) Jha sequence (5) Fermat Family

Journal of Mathematics, 2019

There exist a finite number of Pythagorean triples that have a common leg. In this paper we derive the formulas that generate pairs of primitive Pythagorean triples with common legs and also show the process of how to determine all the primitive and nonprimitive Pythagorean triples for a given leg of a Pythagorean triple.