scholarly journals A note on the diophantine equation x2 = y2 + 3z2

2021 ◽  
Vol 44 (2) ◽  
pp. 201-205
Author(s):  
Abdullah Al Kafi Majumdar ◽  
Abul Kalam Ziauddin Ahmed
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Academy Of Sciences

In the study of 60-degree and 120-degree triangles, one encounters the Diophantine equations of the form x2 = y2 + 3z2. This paper considers the characteristics of the solution of the Diophantine equation. More specifically, it is shown that the equation has solutions of the form x= p = 3n + 1 for some integer n (>0), where p is a prime with 7£ p £199. Journal of Bangladesh Academy of Sciences, Vol. 44, No. 2, 201-205, 2020

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 Quadratic ideals, indefinite quadratic forms and some specific diophantine equations

2018 ◽  
Vol 36 (3) ◽  
pp. 173-192
Author(s):  
Ahmet Tekcan ◽  
Seyma Kutlu
Keyword(s):  
Diophantine Equation ◽  
Quadratic Forms ◽  
Diophantine Equations ◽  
Algebraic Properties ◽  
Integer Solutions ◽  
Indefinite Quadratic

Let $k\geq 1$ be an integer and let $P=k+2,Q=k$ and $D=k^{2}+4$. In this paper, we derived some algebraic properties of quadratic ideals $I_{\gamma}$ and indefinite quadratic forms $F_{\gamma }$ for quadratic irrationals $\gamma$, and then we determine the set of all integer solutions of the Diophantine equation $F_{\gamma }^{\pm k}(x,y)=\pm Q$.

scholarly journals On a class of insoluble binary quadratic diophantine equations

1991 ◽  
Vol 123 ◽  
pp. 141-151 ◽  
Author(s):  
Franz Halter-Koch
Keyword(s):  
Diophantine Equation ◽  
Class Number ◽  
Diophantine Equations ◽  
Number Fields ◽  
Quadratic Number ◽  
Number Problem ◽  
Quadratic Number Fields ◽  
Quadratic Diophantine Equation ◽  
Real Quadratic

The binary quadratic diophantine equationis of interest in the class number problem for real quadratic number fields and was studied in recent years by several authors (see [4], [5], [2] and the literature cited there).

ON EXPONENTIAL DIOPHANTINE EQUATIONS CONTAINING THE EULER QUOTIENT

2014 ◽  
Vol 91 (1) ◽  
pp. 11-18
Author(s):  
NOBUHIRO TERAI
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Euler’S Totient Function ◽  
Exponential Diophantine Equations ◽  
Euler’S Theorem ◽  
Positive Integers

AbstractLet $a$ and $m$ be relatively prime positive integers with $a>1$ and $m>2$. Let ${\it\phi}(m)$ be Euler’s totient function. The quotient $E_{m}(a)=(a^{{\it\phi}(m)}-1)/m$ is called the Euler quotient of $m$ with base $a$. By Euler’s theorem, $E_{m}(a)$ is an integer. In this paper, we consider the Diophantine equation $E_{m}(a)=x^{l}$ in integers $x>1,l>1$. We conjecture that this equation has exactly five solutions $(a,m,x,l)$ except for $(l,m)=(2,3),(2,6)$, and show that if the equation has solutions, then $m=p^{s}$ or $m=2p^{s}$ with $p$ an odd prime and $s\geq 1$.

scholarly journals A Hypothetical Upper Bound on the Heights of the Solutions of a Diophantine Equation with a Finite Number of Solutions

2018 ◽  
Author(s):  
Apoloniusz Tyszka
Keyword(s):  
Positive Integer ◽  
Finite Number ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Diophantine Equations ◽  
Rational Solution ◽  
Infinitely Many Solutions ◽  
Rational Solutions ◽  
Number Of Solutions ◽  
Integer Solutions

Let f ( 1 ) = 1 , and let f ( n + 1 ) = 2 2 f ( n ) for every positive integer n. We consider the following hypothesis: if a system S ⊆ {xi · xj = xk : i, j, k ∈ {1, . . . , n}} ∪ {xi + 1 = xk : i, k ∈{1, . . . , n}} has only finitely many solutions in non-negative integers x1, . . . , xn, then each such solution (x1, . . . , xn) satisfies x1, . . . , xn ≤ f (2n). We prove:   (1) the hypothesisimplies that there exists an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if the solution set is finite; (2) the hypothesis implies that there exists an algorithm for listing the Diophantine equations with infinitely many solutions in non-negative integers; (3) the hypothesis implies that the question whether or not a given Diophantine equation has only finitely many rational solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has a rational solution; (4) the hypothesis implies that the question whether or not a given Diophantine equation has only finitely many integer solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has an integer solution; (5) the hypothesis implies that if a set M ⊆ N has a finite-fold Diophantine representation, then M is computable.

scholarly journals On a class of quartic Diophantine equations

2021 ◽  
Vol 27 (1) ◽  
pp. 1-6
Author(s):  
F. Izadi ◽  
◽  
M. Baghalaghdam ◽  
S. Kosari ◽  
◽  
...  
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Diophantine Equation ◽  
Diophantine Equations ◽  
Nontrivial Solutions ◽  
Natural Numbers ◽  
Positive Rank

In this paper, by using elliptic curves theory, we study the quartic Diophantine equation (DE) { \sum_{i=1}^n a_ix_{i} ^4= \sum_{j=1}^na_j y_{j}^4 }, where a_i and n\geq3 are fixed arbitrary integers. We try to transform this quartic to a cubic elliptic curve of positive rank. We solve the equation for some values of a_i and n=3,4, and find infinitely many nontrivial solutions for each case in natural numbers, and show among other things, how some numbers can be written as sums of three, four, or more biquadrates in two different ways. While our method can be used for solving the equation for n\geq 3, this paper will be restricted to the examples where n=3,4. Finally, we explain how to solve more general cases (n\geq 4) without giving concrete examples to case n\geq 5.

scholarly journals On exponential Diophantine equation 17x +83y = z2 and 29x +71y = z2

2021 ◽  
Vol 2070 (1) ◽  
pp. 012015
Author(s):  
Komon Paisal ◽  
Pailin Chayapham
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Integer Solution ◽  
Negative Integer ◽  
Exponential Diophantine Equation

Abstract This Diophantine is an equation that many researchers are interested in and studied in many form such 3x +5y · 7z = u2, (x+1)k + (x+2)k + … + (2x)k = yn and kax + lby = cz. The extensively studied form is ax + by = cz. In this paper we show that the Diophantine equations 17x +83y = z2 and 29x +71y = z2 has a unique non – negative integer solution (x, y, z) = (1,1,10)

AN APPLICATION OF LINEAR DIOPHANTINE EQUATIONS TO CRYPTOGRAPHY

2021 ◽  
Vol 10 (6) ◽  
pp. 2799-2806
Author(s):  
P. Anuradha Kameswari ◽  
S.S. Sriniasarao ◽  
A. Belay
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Key Exchange ◽  
Infinitely Many Solutions ◽  
Key Exchange Protocol ◽  
Random Solution ◽  
Linear Diophantine Equation ◽  
Linear Diophantine Equations

In this chapter we propose a Key exchange protocol based on a random solution of linear Diophantine equation in n variables, where the considered linear Diophantine equation satisfies the condition for existence of infinitely many solutions. Also the crypt analysis of the protocol is analysed.

Diophantine Equations for Enhanced Security in Watermarking Scheme for Image Authentication

2017 ◽  
pp. 205-229
Author(s):  
Padma T ◽  
Jayashree Nair
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Image Authentication ◽  
Authentication Scheme ◽  
Content Authentication ◽  
Image Content ◽  
Polynomial Time Algorithms ◽  
Mathematical Problems ◽  
Watermarking Scheme

Hard mathematical problems having no polynomial time algorithms to determine a solution are seemly in design of secure cryptosystems. The proposed watermarking system used number theoretic concepts of the hard higher order Diophantine equations for image content authentication scheme with three major phases such as 1) Formation of Diophantine equation; 2) Generation and embedding of dual Watermarks; and 3) Image content authentication and verification of integrity. Quality of the watermarked images, robustness to compression and security are bench-marked with two peer schemes which used dual watermarks.

scholarly journals A Diophantine Equation Associated to X0(5)

2005 ◽  
Vol 8 ◽  
pp. 116-121
Author(s):  
Imin Chen
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Galois Representations ◽  
Point Of View ◽  
Integral Points ◽  
Modular Curve ◽  
Proper Solutions

AbstractSeveral classes of Fermat-type diophantine equations have been successfully resolved using the method of galois representations and modularity. In each case, it is possible to view the proper solutions to the diophantine equation in question as corresponding to suitably defined integral points on a modular curve of level divisible by 2 or 3. Motivated by this point of view, an example of a diophantine equation associated to the modular curve X0(5) is discussed in this paper. The diophantine equation has four terms rather than the usual three terms characteristic of generalized Fermat equations.

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