Fourth power diophantine equations in Gaussian integers

2018 ◽  
Vol 128 (2) ◽  
Author(s):  
Farzali Izadi ◽  
Naghdali Forooshani Rasool ◽  
Amiryousefi Varnousfaderani Amaneh
2005 ◽  
Vol 4 (1) ◽  
pp. 96-100
Author(s):  
A. K. Maran

We know already that the set Of positive integers, which are satisfying the Pythagoras equation Of three variables and four variables cre called Pythagorean triples and quadruples respectively. These cre Diophantine equation OF second power. The all unknowns in this Pythagorean equation have already Seen by mathematicians Euclid and Diophantine. Hcvwever the solution defined by Euclid are Diophantine is also again having unknowns. The only to solve the Diophantine equations wos and error method. Moreover, the trial and error method to obtain these values are not so practical and easy especially for time bound work, since the Diophantine equations are having more than unknown variables.


Author(s):  
M. Gopalan ◽  
S. Vidhyalakshmi ◽  
N. Thiruniraiselvi

Given any two non-zero distinct irrational Gaussian integers such that their product added with either 1 or 4 is a perfect square, an irrational Gaussian Diophantine quadruple ( , ) a0 a1, a2, a3 such that the product of any two members of the set added with either 1 or 4 is a perfect square by employing the non-zero distinct integer solutions of the system of double Diophantine equations. The repetition of the above process leads to the generation of sequences of irrational Gaussian Diophantine quadruples with the given property.


2015 ◽  
Vol 92 (2) ◽  
pp. 187-194
Author(s):  
FARZALI IZADI ◽  
RASOOL FOROOSHANI NAGHDALI ◽  
PETER GEOFF BROWN

In this paper we examine solutions in the Gaussian integers to the Diophantine equation $ax^{4}+by^{4}=cz^{2}$ for different choices of $a,b$ and $c$. Elliptic curve methods are used to show that these equations have a finite number of solutions or have no solution.


2015 ◽  
Vol 3 (2) ◽  
Author(s):  
Jayashree Nair ◽  
T. Padma

This paper describes an authentication scheme that uses Diophantine equations based generation of the secret locations to embed the authentication and recovery watermark in the DWT sub-bands. The security lies in the difficulty of finding a solution to the Diophantine equation. The scheme uses the content invariant features of the image as a self-authenticating watermark and a quantized down sampled approximation of the original image as a recovery watermark for visual authentication, both embedded securely using secret locations generated from solution of the Diophantine equations formed from the PQ sequences. The scheme is mildly robust to Jpeg compression and highly robust to Jpeg2000 compression. The scheme also ensures highly imperceptible watermarked images as the spatio –frequency properties of DWT are utilized to embed the dual watermarks.


Polymers ◽  
2021 ◽  
Vol 13 (14) ◽  
pp. 2286
Author(s):  
Jan Kominek ◽  
Martin Zachar ◽  
Michal Guzej ◽  
Erik Bartuli ◽  
Petr Kotrbacek

Miniaturization of electronic devices leads to new heat dissipation challenges and traditional cooling methods need to be replaced by new better ones. Polymer heat sinks may, thanks to their unique properties, replace standardly used heat sink materials in certain applications, especially in applications with high ambient temperature. Polymers natively dispose of high surface emissivity in comparison with glossy metals. This high emissivity allows a larger amount of heat to be dissipated to the ambient with the fourth power of its absolute surface temperature. This paper shows the change in radiative and convective heat transfer from polymer heat sinks used in different ambient temperatures. Furthermore, the observed polymer heat sinks have differently oriented graphite filler caused by their molding process differences, therefore their thermal conductivity anisotropies and overall cooling efficiencies also differ. Furthermore, it is also shown that a high radiative heat transfer leads to minimizing these cooling efficiency differences between these polymer heat sinks of the same geometry. The measurements were conducted at HEATLAB, Brno University of Technology.


Author(s):  
Debika Banerjee ◽  
Ehud Moshe Baruch ◽  
Daniel Bump

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