scholarly journals On the Exponential Diophantine Equation (132m) + (6r + 1)n = z2

2021 ◽  
Vol 13 (3) ◽  
pp. 845-849
Author(s):  
S. Aggarwal ◽  
S. Kumar
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Abstract Algebra ◽  
Exponential Diophantine Equation ◽  
Whole Numbers ◽  
Whole Number ◽  
Congruence Theory ◽  
Coordinate Geometry

Nowadays, mathematicians are very interested in discovering new and advanced methods for determining the solution of Diophantine equations. Diophantine equations are those equations that have more unknowns than equations. Diophantine equations appear in astronomy, cryptography, abstract algebra, coordinate geometry and trigonometry. Congruence theory plays an important role in finding the solution of some special type Diophantine equations. The absence of any generalized method, which can handle each Diophantine equation, is challenging for researchers. In the present paper, the authors have discussed the existence of the solution of exponential Diophantine equation  (132m) + (6r + 1)n = Z2, where m, n, r, z are whole numbers. Results of the present paper show that the exponential Diophantine equation (132m) + (6r + 1)n = Z2, where m, n, r, z are whole numbers, has no solution in the whole number.

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)

The exponential Diophantine equation x2 + (3n 2 + 1)y = (4n 2 + 1)z

2014 ◽  
Vol 64 (5) ◽  
Author(s):  
Jianping Wang ◽  
Tingting Wang ◽  
Wenpeng Zhang
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Diophantine Equations ◽  
Exponential Diophantine Equation ◽  
Lucas Numbers ◽  
Exponential Diophantine Equations ◽  
Primitive Divisors ◽  
Integer Solutions

AbstractLet n be a positive integer. In this paper, using the results on the existence of primitive divisors of Lucas numbers and some properties of quadratic and exponential diophantine equations, we prove that if n ≡ 3 (mod 6), then the equation x 2 + (3n 2 + 1)y = (4n 2 + 1)z has only the positive integer solutions (x, y, z) = (n, 1, 1) and (8n 3 + 3n, 1, 3).

scholarly journals A note on the exponential Diophantine equation (A^2n)^x+(B^2n)^y=((A^2+B^2)n)^z

2020 ◽  
Vol 55 (2) ◽  
pp. 195-201
Author(s):  
Maohua Le ◽  
◽  
Gökhan Soydan ◽  
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Diophantine Equations ◽  
Exponential Diophantine Equation ◽  
Positive Integer Solution ◽  
Exponential Diophantine Equations ◽  
Integer Solutions ◽  
Positive Integers ◽  
Upper Bound For Solutions

Let A, B be positive integers such that min{A,B}>1, gcd(A,B) = 1 and 2|B. In this paper, using an upper bound for solutions of ternary purely exponential Diophantine equations due to R. Scott and R. Styer, we prove that, for any positive integer n, if A >B3/8, then the equation (A2 n)x + (B2 n)y = ((A2 + B2)n)z has no positive integer solutions (x,y,z) with x > z > y; if B>A3/6, then it has no solutions (x,y,z) with y>z>x. Thus, combining the above conclusion with some existing results, we can deduce that, for any positive integer n, if B ≡ 2 (mod 4) and A >B3/8, then this equation has only the positive integer solution (x,y,z)=(1,1,1).

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 Solving an exponential Diophantine equation

2010 ◽  
Vol 144 (4) ◽  
pp. 315-322
Author(s):  
Mario Huicochea
Keyword(s):  
Diophantine Equation ◽  
Exponential Diophantine Equation

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 $F_{\gamma }^{\pm k}(x,y)=\pm Q$.

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