On the Diophantine Equation x 2 + 2 α 5 β 13 γ = y n

Lecture Notes in Computer Science - Algorithmic Number Theory ◽

10.1007/978-3-540-79456-1_29 ◽

2008 ◽

pp. 430-442 ◽

Cited By ~ 4

Author(s):

Edray Goins ◽

Florian Luca ◽

Alain Togbé

Keyword(s):

Diophantine Equation

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An Elementary Approach to the Diophantine Equation $ax^m + by^n = z^r$ Using Center of Mass

Missouri Journal of Mathematical Sciences ◽

10.35834/mjms/1513306825 ◽

2017 ◽

Vol 29 (2) ◽

pp. 115-124

Author(s):

Amir M. Rahimi

Keyword(s):

Diophantine Equation ◽

Center Of Mass ◽

Elementary Approach

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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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On the Ternary Exponential Diophantine Equation Equating a Perfect Power and Sum of Products of Consecutive Integers

Mathematics ◽

10.3390/math9151813 ◽

2021 ◽

Vol 9 (15) ◽

pp. 1813

Author(s):

S. Subburam ◽

Lewis Nkenyereye ◽

N. Anbazhagan ◽

S. Amutha ◽

M. Kameswari ◽

...

Keyword(s):

Positive Integer ◽

Diophantine Equation ◽

Upper Bound ◽

Exponential Diophantine Equation ◽

Research Article ◽

All Solutions ◽

Sum Of Products ◽

Consecutive Integers ◽

Sums Of Products ◽

Positive Integers

Consider the Diophantine equation yn=x+x(x+1)+⋯+x(x+1)⋯(x+k), where x, y, n, and k are integers. In 2016, a research article, entitled – ’power values of sums of products of consecutive integers’, primarily proved the inequality n= 19,736 to obtain all solutions (x,y,n) of the equation for the fixed positive integers k≤10. In this paper, we improve the bound as n≤ 10,000 for the same case k≤10, and for any fixed general positive integer k, we give an upper bound depending only on k for n.

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