A group of Pythagorean triples using the inradius

Pythagorean triples are triples of integers (a, b, c) satisfying the equation a2 + b2 = c2. For the purpose of this paper, we will take a, b and c to be positive, unless otherwise stated. Then, of course, it follows that a triple represents the lengths of sides of a right triangle. Also, for the purpose of this paper, we will consider the triples (a, b, c) and (b, a, c) to be distinct, even though they represent the same right triangle. A primitive Pythagorean triple is one for which a, b and c are relatively prime.

Some Results on a Generalized Version of Congruent Numbers

This paper aims to construct a new formula that generates a generalized version of congruent numbers based on a generalized version of Pythagorean triples. Here, an elliptic curve equation is constructed from the derived generalized version of Pythagorean triples and congruent numbers and gives some new results.Keywords: Pythagorean triple, congruent number, elliptic curve equation. AbstrakArtikel ini bertujuan untuk mengkonstruksi formula baru yang membangun versi yang lebih umum dari bilangan-bilangan kongruen berdasarkan versi triple Pythagoras yang diperumum. Di sini, akan dikonstruksi suatu persamaan kurva eliptik dari triple Pythagoras dan bilangan-bilangan kongruen dalam versi yang diperumum untuk menghasilkan hasil-hasil yang baru.Kata kunci: triple Phytagoras, bilangan kongruen, persamaan kurva eliptik.2010 Mathematics subject classification: 11A07, 11A41, 11D45, 11G07.

A note on Jeśmanowicz’ conjecture for non-primitive Pythagorean triples

Let \((a, b, c)\) be a primitive Pythagorean triple parameterized as \(a=u^2-v^2, b=2uv, c=u^2+v^2\), where \(u>v>0\) are co-prime and not of the same parity. In 1956, L. Jesmanowicz conjectured that for any positive integer \(n\), the Diophantine equation \((an)^x+(bn)^y=(cn)^z\) has only the positive integer solution \((x,y,z)=(2,2,2)\). In this connection we call a positive integer solution \((x,y,z)\ne (2,2,2)\) with \(n>1\) exceptional. In 1999 M.-H. Le gave necessary conditions for the existence of exceptional solutions which were refined recently by H. Yang and R.-Q. Fu. In this paper we give a unified simple proof of the theorem of Le-Yang-Fu. Next we give necessary conditions for the existence of exceptional solutions in the case \(v=2,\ u\) is an odd prime. As an application we show the truth of the Jesmanowicz conjecture for all prime values \(u < 100\).

Golden Ratio

This paper introduces the unique geometric features of 1:2: right triangle, which is observed to be the quintessential form of Golden Ratio (φ). The 1:2: triangle, with all its peculiar geometric attributes described herein, turns out to be the real ‘Golden Ratio Triangle’ in every sense of the term. This special right triangle also reveals the fundamental Pi:Phi (π:φ) correlation, in terms of precise geometric ratios, with an extreme level of precision. Further, this 1:2: triangle is found to have a classical geometric relationship with 3-4-5 Pythagorean triple. The perfect complementary relationship between1:2: triangle and 3-4-5 triangle not only unveils several new aspects of Golden Ratio, but it also imparts the most accurate π:φ correlation, which is firmly premised upon the classical geometric principles. Moreover, this paper introduces the concept of special right triangles; those provide the generalised geometric substantiation of all Metallic Means.

SOME NOTES ON A GENERALIZED VERSION OF PYTHAGOREAN TRIPLES

A Pythagorean triple is a set of three positive integers a, b and c that satisfy the Diophantine equation a^2+b^2=c^2. The triple is said to be primitive if gcd(a, b, c)=1 and each pair of integers and are relatively prime, otherwise known as non-primitive. In this paper, the generalized version of the formula that generates primitive and non-primitive Pythagorean triples that depends on two positive integers k and n, that is, P_T=(a(k, n), b(k, n), c(k, n)) were constructed. Further, we determined the values of k and n that generates primitive Pythagorean triples and give some important results.

Pythagorean Triples with Common Sides

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.

Everything Flows: Continuous Micro-Flow for Pharmaceutical Production

The pre-Socratic philosophers made the first honest attempt, at least in the western world, to describe natural phenomena in a rudimentary scientific manner and to exploit those for technological application [1]. Pythagoras of Samos (570–495 BC) was an Ionian Greek philosopher and the first to actually call himself a “philosopher”. He was credited with many mathematical and scientific discoveries, including the Pythagorean theorem, Pythagorean tuning, the five regular solids, the theory of proportions, and the sphericity of the Earth. The Pythagorean triple is also well-known. Heraclitus of Ephesus (535–475 BC) was famous for his insistence on ever-present change as the fundamental essence of the universe, as stated in the famous saying“panta rhei”—everything flows.