منابع مشابه
The Congruent Number Problem
We discuss a famous problem about right triangles with rational side lengths. This elementarysounding problem is still not completely solved; the last remaining step involves the Birch and Swinnerton-Dyer conjecture, which is one of the most important open problems in number theory (right up there with the Riemann hypothesis). 6.
متن کاملSome Variants of the Congruent Number Problem III
for square-free integers m, and we have obtained that the special value L(Em, 1) of L-function for Em is described in terms of positive definite ternary quadratic forms. It is an analogue result of Tunnell’s [Tun] for the classical congruent number problem. In this paper, we use a result of Gauss on ternary quadratic forms to give more useful criterion for non-2π/3and non-2π/3-congruentness. So...
متن کاملCongruent Number Theta Coefficients to 1012
We report on a computation of congruent numbers, which subject to the Birch and Swinnerton-Dyer conjecture is an accurate list up to 10. The computation involves multiplying large theta series as per Tunnell (1983). The first method, which we describe in some detail, uses a multimodular disk based technique for multiplying polynomials out-of-core which minimises expensive disk access by keeping...
متن کاملIntegral points on congruent number curves
We provide a precise description of the integer points on elliptic curves of the shape y2 = x3 − N2x, where N = 2apb for prime p. By way of example, if p ≡ ±3 (mod 8) and p > 29, we show that all such points necessarily have y = 0. Our proofs rely upon lower bounds for linear forms in logarithms, a variety of old and new results on quartic and other Diophantine equations, and a large amount of ...
متن کاملGauss’s 2f1 Hypergeometric Function and the Congruent Number Elliptic Curve
Gauss’s hypergeometric function gives a modular parameterization of period integrals of elliptic curves in Legendre normal form E(λ) : y = x(x− 1)(x− λ). We study a modular function which “measures” the variation of periods for the isomorphic curves E(λ) and E ( λ λ−1 ) , and we show that it padically “interpolates” the cusp form for the “congruent number” curve E(2), the case where these pairs...
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ژورنال
عنوان ژورنال: Pure and Applied Mathematics Quarterly
سال: 2005
ISSN: 1558-8599,1558-8602
DOI: 10.4310/pamq.2005.v1.n1.a2