Microcomputer-assisted mathematics: How to Solve the Diophantine Equation x2 + y2 = z3 With Only One Literal Symbol

1985 ◽  
Vol 78 (9) ◽  
pp. 708-711
Author(s):  
Peter Flusser
Keyword(s):  
Diophantine Equation ◽  
Great Mathematician

Little is known of the life of the great mathematician Diophantus. We know that he lived in Alexandria, Egypt. His name was Greek, he wrote in Greek, and his heritage was mostly Greek, with perhaps some Persian influence. He may have been Egyptian.

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

3223. The Diophantine Equation x 2 +y 2 +z 2 = u 2

1969 ◽  
Vol 53 (384) ◽  
pp. 176
Author(s):  
Bernard Lawrence
Keyword(s):  
Diophantine Equation

scholarly journals Note on a Fermat-type diophantine equation

2003 ◽  
Vol 99 (1) ◽  
pp. 29-35
Author(s):  
Sankar Sitaraman
Keyword(s):  
Diophantine Equation
Sign in / Sign up

Export Citation Format

Share Document