scholarly journals Quadratic Diophantine Equations, the Class Number, and the Mass Formula

2016 ◽  
pp. 87-106
Author(s):  
Goro Shimura
Keyword(s):  
Class Number ◽  
Mass Formula ◽  
Diophantine Equations

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).

scholarly journals NOTE ON THE DIVISIBILITY OF THE CLASS NUMBER OF CERTAIN IMAGINARY QUADRATIC FIELDS

2009 ◽  
Vol 51 (1) ◽  
pp. 187-191 ◽  
Author(s):  
YASUHIRO KISHI
Keyword(s):  
Class Number ◽  
Diophantine Equations ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Positive Integers

AbstractWe prove that the class number of the imaginary quadratic field $\Q(\sqrt{2^{2k}-3^n})$ is divisible by n for any positive integers k and n with 22k < 3n, by using Y. Bugeaud and T. N. Shorey's result on Diophantine equations.

scholarly journals The fundamental unit and bounds for class numbers of real quadratic fields

1991 ◽  
Vol 124 ◽  
pp. 181-197 ◽  
Author(s):  
Hideo Yokoi
Keyword(s):  
Class Number ◽  
Continued Fraction ◽  
Diophantine Equations ◽  
Fundamental Unit ◽  
Quadratic Fields ◽  
Class Numbers ◽  
Imaginary Quadratic Fields ◽  
Real Quadratic Fields ◽  
Real Quadratic

Although class number one problem for imaginary quadratic fields was solved in 1966 by A. Baker [3] and by H. M. Stark [25] independently, the problem for real quadratic fields remains still unsettled. However, since papers by Ankeny–Chowla–Hasse [2] and H. Hasse [9], many papers concerning this problem or giving estimate for class numbers of real quadratic fields from below have appeared. There are three methods used there, namely the first is related with quadratic diophantine equations ([2], [9], [27, 28, 29, 31], [17]), and the second is related with continued fraction expantions ([8], [4], [16], [14], [18]).

scholarly journals On the System of Diophantine Equationsx2-6y2=-5andx=az2-b

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Silan Zhang ◽  
Jianhua Chen ◽  
Hao Hu
Keyword(s):  
Number Theory ◽  
Algebraic Number ◽  
Class Number ◽  
Diophantine Equations ◽  
Algebraic Number Theory ◽  
The Family ◽  
Integral Solutions

Mignotte and Pethö used the Siegel-Baker method to find all the integral solutions(x,y,z)of the system of Diophantine equationsx2-6y2=-5andx=2z2-1. In this paper, we extend this result and put forward a generalized method which can completely solve the family of systems of Diophantine equationsx2-6y2=-5andx=az2-bfor each pair of integral parametersa,b. The proof utilizes algebraic number theory andp-adic analysis which successfully avoid discussing the class number and factoring the ideals.

Note on class number parity of an abelian field of prime conductor, II

2019 ◽  
Vol 42 (1) ◽  
pp. 99-110 ◽  
Author(s):  
Humio Ichimura
Keyword(s):  
Class Number ◽  
Abelian Field

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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