Higher Descent on Pell Conics. Iii. the First 2-descent
نویسنده
چکیده
In [Lem2003b] we have sketched the historical development of problems related to Legendre’s equations ar−bs = 1 and the associated Pell equation x−dy = 1 with d = ab. In [Lem2003c] we discussed certain “non-standard” ideas to solve the Pell equation. Now we move from the historical to the modern part: below we will describe the theory of the first 2-descent on Pell conics and explain its connections to some of the results described in [Lem2003b], leaving the theory of the second 2-descent and its relations to results from [Lem2003c] to another occasion. As everyone familiar with the basic arithmetic of elliptic curves will notice, many of the results (e.g. those on heights) presented here are special cases of more general theorems.
منابع مشابه
Higher Descent on Pell Conics. I. from Legendre to Selmer
The theory of Pell’s equation has a long history, as can be seen from the huge amount of references collected in Dickson [Dic1920], from the two books on its history by Konen [Kon1901] and Whitford [Whi1912], or from the books by Weber [Web1939], Walfisz [Wal1952], Faisant [Fai1991], and Barbeau [Bar2003]. For the better part of the last few centuries, the continued fractions method was the und...
متن کاملConics - a Poor Man’s Elliptic Curves
Introduction 2 1. The Group Law on Pell Conics and Elliptic Curves 2 1.1. Group Law on Conics 2 1.2. Group Law on Elliptic curves 3 2. The Group Structure 3 2.1. Finite Fields 3 2.2. p-adic Numbers 3 2.3. Integral and Rational Points 4 3. Applications 4 3.1. Primality Tests 4 3.2. Factorization Methods 5 4. 2-Descent 5 4.1. Selmer and Tate-Shafarevich Group 5 4.2. Heights 6 5. Analytic Methods ...
متن کامل1 8 N ov 2 00 3 HIGHER DESCENT ON PELL CONICS . II . TWO CENTURIES OF MISSED OPPORTUNITIES
It was already observed by Euler [Eul1773] that the method of continued fractions occasionally requires a lot of tedious calculations, and even Fermat knew – as can be seen from the examples he chose to challenge the English mathematicians – a few examples with large solutions. To save work, Euler suggested a completely different method, which allows to compute even very large solutions of cert...
متن کاملPrevalence of Allergies and Food Intolerance: a Comparison Between the Persian Community and Canadians of European Descent
Background: The prevalence of reported allergic diseases is higher in North-America than in other countries around the world. Further, certain types of allergies are more common in some geographic regions or amongst different ethnic populations. This phenomenon has not been well studied amongst the Persian population in the Canadian context; therefore, the current study aimed to compare the pre...
متن کاملA Free Line Search Steepest Descent Method for Solving Unconstrained Optimization Problems
In this paper, we solve unconstrained optimization problem using a free line search steepest descent method. First, we propose a double parameter scaled quasi Newton formula for calculating an approximation of the Hessian matrix. The approximation obtained from this formula is a positive definite matrix that is satisfied in the standard secant relation. We also show that the largest eigen value...
متن کامل