The $${\theta }$$-Congruent Number Elliptic Curves via Fermat-type Algorithms

2020 ◽  
Author(s):  
Sajad Salami ◽  
Arman Shamsi Zargar
Keyword(s):  
Elliptic Curves ◽  
Congruent Number

Congruent Number Elliptic Curves with Rank at Least Three

2010 ◽  
Vol 53 (4) ◽  
pp. 661-666 ◽  
Author(s):  
Jennifer A. Johnstone ◽  
Blair K. Spearman
Keyword(s):  
Elliptic Curves ◽  
Infinite Family ◽  
Congruent Number

AbstractWe give an infinite family of congruent number elliptic curves each with rank at least three.

scholarly journals Right triangles with algebraic sides and elliptic curves over number fields

2009 ◽  
Vol 59 (3) ◽  
Author(s):  
E. Girondo ◽  
G. González-Diez ◽  
E. González-Jiménez ◽  
R. Steuding ◽  
J. Steuding
Keyword(s):  
Positive Integer ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Infinite Order ◽  
Number Fields ◽  
Explicit Construction ◽  
Congruent Number ◽  
Number Problem ◽  
Weil Group ◽  
Congruent Number Problem

AbstractGiven any positive integer n, we prove the existence of infinitely many right triangles with area n and side lengths in certain number fields. This generalizes the famous congruent number problem. The proof allows the explicit construction of these triangles; for this purpose we find for any positive integer n an explicit cubic number field ℚ(λ) (depending on n) and an explicit point P λ of infinite order in the Mordell-Weil group of the elliptic curve Y 2 = X 3 − n 2 X over ℚ(λ).

scholarly journals On high rank $\pi/3$ and $2\pi/3$-congruent number elliptic curves

2014 ◽  
Vol 44 (6) ◽  
pp. 1867-1880 ◽  
Author(s):  
A.S. Janfada ◽  
S. Salami ◽  
A. Dujella ◽  
J.C. Peral
Keyword(s):  
Elliptic Curves ◽  
High Rank ◽  
Congruent Number

INTEGRAL POINTS ON CONGRUENT NUMBER CURVES

2013 ◽  
Vol 09 (06) ◽  
pp. 1619-1640 ◽  
Author(s):  
MICHAEL A. BENNETT
Keyword(s):  
Elliptic Curves ◽  
Lower Bounds ◽  
Diophantine Equations ◽  
Linear Forms In Logarithms ◽  
Precise Description ◽  
Integral Points ◽  
Linear Forms ◽  
Integer Points ◽  
Congruent Number

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 (non-trivial) computation.

scholarly journals ON A NONCRITICAL SYMMETRIC SQUARE -VALUE OF THE CONGRUENT NUMBER ELLIPTIC CURVES

2019 ◽  
Vol 101 (1) ◽  
pp. 13-22
Author(s):  
DETCHAT SAMART
Keyword(s):  
Elliptic Curves ◽  
Simple Proof ◽  
Degree Zero ◽  
Torsion Points ◽  
Zero Divisors ◽  
Congruent Number ◽  
Symmetric Square

The congruent number elliptic curves are defined by $E_{d}:y^{2}=x^{3}-d^{2}x$, where $d\in \mathbb{N}$. We give a simple proof of a formula for $L(\operatorname{Sym}^{2}(E_{d}),3)$ in terms of the determinant of the elliptic trilogarithm evaluated at some degree zero divisors supported on the torsion points on $E_{d}(\overline{\mathbb{Q}})$.

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