Dem’janenko’s Theorem on Jeśmanowicz’ Conjecture Concerning Pythagorean Triples Revisited

2021 ◽  
Author(s):  
Yasutsugu Fujita ◽  
Maohua Le
Keyword(s):  
Pythagorean Triples

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

85.26 Another Method for Pythagorean Triples

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

scholarly journals Pythagorean Triples in Cryptography and Associated Networks

2019 ◽  
Vol 8 (12) ◽  
pp. 832-837
Keyword(s):  
Plain Text ◽  
Data Preservation ◽  
Pythagorean Triples ◽  
Increasing Demand ◽  
Secured Data

Any organization is obliged to ensure secrecy of data from hacking criminals complying with the increasing demand for secured data. So data preservation is indispensablethrough cryptographic methods. It is adoptedseveralreal life applications such as ecommerce, In this paper we indicated a procedure for generating different Pythagorean triples and with the help of C++ coding developed a mechanism for both encoding and decoding of a plain text in English alphabets. We also demonstrated them with illustrative examples

THE SHUFFLE VARIANT OF JEŚMANOWICZ’ CONJECTURE CONCERNING PYTHAGOREAN TRIPLES

2011 ◽  
Vol 90 (3) ◽  
pp. 355-370
Author(s):  
TAKAFUMI MIYAZAKI
Keyword(s):  
Unique Solution ◽  
Pythagorean Triples ◽  
Pythagorean Triple ◽  
Positive Integers

AbstractLet (a,b,c) be a primitive Pythagorean triple such that b is even. In 1956, Jeśmanowicz conjectured that the equation ax+by=cz has the unique solution (x,y,z)=(2,2,2) in the positive integers. This is one of the most famous unsolved problems on Pythagorean triples. In this paper we propose a similar problem (which we call the shuffle variant of Jeśmanowicz’ problem). Our problem states that the equation cx+by=az with x,y and z positive integers has the unique solution (x,y,z)=(1,1,2) if c=b+1 and has no solutions if c>b+1 . We prove that the shuffle variant of the Jeśmanowicz problem is true if c≡1 mod b.

79.60 Pythagorean Triples

1995 ◽  
Vol 79 (486) ◽  
pp. 574
Author(s):  
Shawn Glasco
Keyword(s):  
Pythagorean Triples

SOME NOTES ON A GENERALIZED VERSION OF PYTHAGOREAN TRIPLES

2020 ◽  
Vol 4 (2) ◽  
pp. 103
Author(s):  
Leomarich F Casinillo ◽  
Emily L Casinillo
Keyword(s):  
Diophantine Equation ◽  
Pythagorean Triples ◽  
Pythagorean Triple ◽  
Positive Integers

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.

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