Exceptional cases of Terai’s conjecture on Diophantine equations

2011 ◽  
Author(s):  
Takafumi Miyazaki ◽  
Masaaki Amou ◽  
Masanori Katsurada
Keyword(s):  
Diophantine Equations ◽  
Terai’S Conjecture

TERAI'S CONJECTURE ON EXPONENTIAL DIOPHANTINE EQUATIONS

2011 ◽  
Vol 07 (04) ◽  
pp. 981-999 ◽  
Author(s):  
TAKAFUMI MIYAZAKI
Keyword(s):  
Diophantine Equations ◽  
Exponential Diophantine Equations ◽  
Terai’S Conjecture ◽  
Positive Integers ◽  
Integral Solutions

Let a, b, c be relatively prime positive integers such that ap + bq = cr with fixed integers p, q, r ≥ 2. Terai conjectured that the equation ax + by = cz has no positive integral solutions other than (x, y, z) = (p, q, r) except for specific cases. Most known results on this conjecture concern the case where p = q = 2 and either r = 2 or odd r ≥3. In this paper, we consider the case where p = q = 2 and r > 2 is even, and partially verify Terai's conjecture.

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.

Eight Diophantine Equations

1966 ◽  
Vol s3-16 (1) ◽  
pp. 153-166 ◽  
Author(s):  
J. H. E. Cohn
Keyword(s):  
Diophantine Equations
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