An effective estimation of the bounds for the sizes of solutions of a class of Diophantine equations of the norm form

1983 ◽  
Vol 33 (6) ◽  
pp. 411-414
Author(s):  
S. V. Kotov
Keyword(s):  
Diophantine Equations ◽  
Norm Form

Continued Fractions, Jacobi Symbols, and Quadratic Diophantine Equations

2000 ◽  
Vol 43 (2) ◽  
pp. 218-225 ◽  
Author(s):  
R. A. Mollin ◽  
A. J. van der Poorten
Keyword(s):  
Continued Fractions ◽  
Diophantine Equations ◽  
Norm Form

AbstractThe results herein continue observations on norm form equations and continued fractions begun and continued in the works [1]−[3], and [5]−[6].

scholarly journals MINOR ARC MOMENTS OF WEYL SUMS

2012 ◽  
Vol 55 (1) ◽  
pp. 97-113 ◽  
Author(s):  
M. P. HARVEY
Keyword(s):  
Diophantine Equations ◽  
Diagonal Form ◽  
Norm Form ◽  
Goldbach Problem ◽  
Weyl Sum ◽  
Diophantine Problems

AbstractWe obtain an improved bound for the 2k-th moment of a degree k Weyl sum, restricted to a set of minor arcs, when k is small. We then present some applications of this bound to some Diophantine problems, including a case of the Waring–Goldbach problem, and a particular family of Diophantine equations defined as the sum of a norm form and a diagonal form.

scholarly journals On the finiteness of solutions for polynomial-factorial Diophantine equations

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Wataru Takeda
Keyword(s):  
Diophantine Equations ◽  
Field Extension ◽  
Infinite Subset ◽  
Norm Form ◽  
Factorial Function ◽  
Integer Solutions

AbstractWe study the Diophantine equations obtained by equating a polynomial and the factorial function, and prove the finiteness of integer solutions under certain conditions. For example, we show that there exist only finitely many l such that {l!} is represented by {N_{A}(x)}, where {N_{A}} is a norm form constructed from the field norm of a field extension {K/\mathbf{Q}}. We also deal with the equation {N_{A}(x)=l!_{S}}, where {l!_{S}} is the Bhargava factorial. In this paper, we also show that the Oesterlé–Masser conjecture implies that for any infinite subset S of {\mathbf{Z}} and for any polynomial {P(x)\in\mathbf{Z}[x]} of degree 2 or more the equation {P(x)=l!_{S}} has only finitely many solutions {(x,l)}. For some special infinite subsets S of {\mathbf{Z}}, we can show the finiteness of solutions for the equation {P(x)=l!_{S}} unconditionally.

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.

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