On quartic Diophantine equations with trivial solutions in the Gaussian integers

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.

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SOME QUARTIC DIOPHANTINE EQUATIONS IN THE GAUSSIAN INTEGERS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972715000465 ◽

2015 ◽

Vol 92 (2) ◽

pp. 187-194

Author(s):

FARZALI IZADI ◽

RASOOL FOROOSHANI NAGHDALI ◽

PETER GEOFF BROWN

Keyword(s):

Finite Number ◽

Elliptic Curve ◽

Diophantine Equation ◽

Diophantine Equations ◽

Number Of Solutions ◽

Gaussian Integers

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.

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Properties of Some Diophantine Equations

SSRN Electronic Journal ◽

10.2139/ssrn.3527801 ◽

2020 ◽

Author(s):

Viktor Grechyn

Keyword(s):

Diophantine Equations

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Secure Watermarking using Diophantine Equations for Authentication and Recovery

Journal of Network and Information Security ◽

10.21863/jnis/2015.3.2.006 ◽

2015 ◽

Vol 3 (2) ◽

Author(s):

Jayashree Nair ◽

T. Padma

Keyword(s):

Diophantine Equation ◽

Diophantine Equations ◽

Jpeg Compression ◽

Authentication Scheme ◽

Original Image ◽

Invariant Features ◽

Jpeg2000 Compression ◽

Frequency Properties ◽

Visual Authentication

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.

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