scholarly journals The negative Pell equation and Pythagorean triples

2000 ◽  
Vol 76 (6) ◽  
pp. 91-94 ◽  
Author(s):  
Aleksander Grytczk ◽  
Florian Luca ◽  
Marek Wójtowicz
Keyword(s):  
Pell Equation ◽  
Pythagorean Triples

A kaleidoscope of solutions for a Diophantine system

2010 ◽  
Vol 94 (529) ◽  
pp. 42-50
Author(s):  
Juan Pla
Keyword(s):  
Analogous Problem ◽  
Pell Equation ◽  
Pythagorean Triples ◽  
Diophantine System ◽  
Consecutive Integers ◽  
Image Position

A classical exercise in recreational mathematics is to find Pythagorean triples such that the legs are consecutive integers. It is equivalent to solve the Pell equation with k = 2. In this case it provides all the solutions (see [1] for details). But to obtain all the solutions of a Diophantine system in one stroke is rather exceptional. Actually this note will show that the analogous problem of finding four integers A, B, C and D such that

On the Origin of Four Pi in Physics

2016 ◽  
pp. 3994-4013
Author(s):  
Aaron Hanken
Keyword(s):  
Black Hole ◽  
Surface Area ◽  
Single Particles ◽  
Pythagorean Triples ◽  
Free Particles ◽  
Close Relationship ◽  
Dimensional Parameters ◽  
Gaussian Surface ◽  
Planck Energy ◽  
The Universe

We find the highest symmetry between the fields intrinsic to free particles (free particles having only mass, charge and spin), and show these fields symmetries and their close relationship to force and entropy. The Boltzmann Constant is equal to the natural entropy, in that it is The Planck Energy over The Planck Temperature. This completes a needed symmetry in The Bekenstein-Hawking Entropy. Upon substitution of Planck Units into The Schwarzschild Radius, we find that the mass and radius of any black hole define both the gravitational constant and the natural force. We find that the Gaussian Surface area about a particle is equal to the surface area of an equally massed black hole if we define the gravitational field of that particle to be the quotient of The Planck Force and the particles mass. By these simple substitutions we find that gravity is quantized in units of surface entropy. We also find Pythagorean Triples are resting within the dimensional parameters of Special Relativity, and show this to be the dimensional aspects of single particles observing one another, coupled with the intrinsic Hubble nature of the universe.

Proof Without Words: Parametric Representation of Primitive Pythagorean Triples

1996 ◽  
Vol 69 (3) ◽  
pp. 189-189
Author(s):  
Raymond A. Beauregard ◽  
E. R. Suryanarayan
Keyword(s):  
Parametric Representation ◽  
Pythagorean Triples

The Pell equation over the Fibonacci ∘-ring

2008 ◽  
Vol 150 (3) ◽  
pp. 2084-2095
Author(s):  
V. G. Zhuravlev
Keyword(s):  
Pell Equation

85.26 Another Method for Pythagorean Triples

2001 ◽  
Vol 85 (503) ◽  
pp. 273
Author(s):  
Hassan A. Shah Ali
Keyword(s):  
Pythagorean Triples

Cryptanalysis of the RSA Variant Based on Cubic Pell Equation

2021 ◽  
Author(s):  
Mengce Zheng ◽  
Noboru Kunihiro ◽  
Yuanzhi Yao
Keyword(s):  
Pell Equation

scholarly journals Solution of certain Pell equations

2018 ◽  
Vol 11 (04) ◽  
pp. 1850056 ◽  
Author(s):  
Zahid Raza ◽  
Hafsa Masood Malik
Keyword(s):  
Positive Integer ◽  
Fundamental Solution ◽  
Positive Solutions ◽  
Continued Fraction ◽  
Pell Equation ◽  
Lucas Sequences ◽  
Pell Equations ◽  
Integer Solutions ◽  
Positive Integers

Let [Formula: see text] be any positive integers such that [Formula: see text] and [Formula: see text] is a square free positive integer of the form [Formula: see text] where [Formula: see text] and [Formula: see text] The main focus of this paper is to find the fundamental solution of the equation [Formula: see text] with the help of the continued fraction of [Formula: see text] We also obtain all the positive solutions of the equations [Formula: see text] and [Formula: see text] by means of the Fibonacci and Lucas sequences.Furthermore, in this work, we derive some algebraic relations on the Pell form [Formula: see text] including cycle, proper cycle, reduction and proper automorphism of it. We also determine the integer solutions of the Pell equation [Formula: see text] in terms of [Formula: see text] We extend all the results of the papers [3, 10, 27, 37].

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