Diophantine Equations Connected with the Cubic Fermat Equation

J. D. Swift

doi:10.2307/2304770

Several Diophantine Equations in Some Rings of Integers of Quadratic Imaginary Fields

Algebra Colloquium ◽

10.1142/s1005386707000612 ◽

2007 ◽

Vol 14 (04) ◽

pp. 661-668 ◽

Cited By ~ 2

Author(s):

Kejian Xu ◽

Yongliang Wang

Keyword(s):

Diophantine Equation ◽

Diophantine Equations ◽

Ring Of Integers ◽

Rings Of Integers ◽

Quadratic Imaginary Fields ◽

Fermat Equation

In this paper, it is proved that the Diophantine equation x4-y4 =z2 has no non-trivial coprime solutions in the rings of integers of quadratic imaginary fields [Formula: see text] for d=11, 19, 43, 67, 163, which implies that the Fermat equation x4+y4 =z4 has no non-trivial solutions in these fields either. Then all the solutions of the Pocklington equation x4-x2y2+y4 =(-1)σz2 (σ =0 or 1) in the ring of integers of [Formula: see text] are determined, and as an application, the result is applied to K2 of a field.

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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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