ON AN ELEMENTARY APPROACH TO THE LEBESGUE–NAGELL EQUATION

2005 ◽  
Vol 01 (04) ◽  
pp. 553-561
Author(s):  
R. A. MOLLIN
Keyword(s):  
Diophantine Equation ◽  
Elementary Solution ◽  
Elementary Approach

We discuss the feasibility of an elementary solution to the Diophantine equation of the form x2 + D = yn, where D > 1, n ≥ 3 and x > 0, called the Lebesgue–Nagell equation, which has recently been solved for 1 ≤ D ≤ 100 in [1].

Secure Watermarking using Diophantine Equations for Authentication and Recovery

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.

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

Mathematics ◽  
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.

scholarly journals On the Diophantine Equation $$cx^2+p^{2m}=4y^n$$

2021 ◽  
Vol 76 (2) ◽  
Author(s):  
K. Chakraborty ◽  
A. Hoque ◽  
K. Srinivas
Keyword(s):  
Diophantine Equation

On the Diophantine equation (( c + 1) m 2 + 1) x + ( cm 2

2018 ◽  
Vol 42 (5) ◽  
pp. 2690-2698 ◽  
Author(s):  
Elif KIZILDERE ◽  
Takafumi MIYAZAKI ◽  
Gökhan SOYDAN
Keyword(s):  
Diophantine Equation

Old and New Results for First Order Periodic ODEs without Uniqueness: a Comprehensive Study by Lower and Upper Solutions

2004 ◽  
Vol 4 (3) ◽  
Author(s):  
Franco Obersnel ◽  
Pierpaolo Omari
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Lower And Upper Solutions ◽  
Elementary Approach ◽  
Qualitative Properties ◽  
First Order ◽  
The Past ◽  
The Cauchy Problem ◽  
Qualitative Properties Of Solutions ◽  
Comprehensive Study

AbstractAn elementary approach, based on a systematic use of lower and upper solutions, is employed to detect the qualitative properties of solutions of first order scalar periodic ordinary differential equations. This study is carried out in the Carathéodory setting, avoiding any uniqueness assumption, in the future or in the past, for the Cauchy problem. Various classical and recent results are recovered and generalized.

Elementary solution of Classical Spin-Glass models

1986 ◽  
Vol 65 (1) ◽  
pp. 53-63 ◽  
Author(s):  
J. L. van Hemmen ◽  
D. Grensing ◽  
A. Huber ◽  
R. Kühn
Keyword(s):  
Spin Glass ◽  
Elementary Solution ◽  
Classical Spin
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