scholarly journals The congruent number problem

2021 ◽  
Author(s):  
Jan Feliksiak
Keyword(s):  
Finite Number ◽  
Natural Number ◽  
Mathematical Problem ◽  
Straightforward Manner ◽  
Congruent Number ◽  
Number Problem ◽  
Congruent Number Problem ◽  
2D Array

The congruent number problem is the oldest unsolved major mathematical problem to date. The problem aiming to determine whether or not some given integer n is congruent, which corresponds to a Pythagorean triangle with integer sides, can be settled in a finite number of steps. However, once we permit the triangles to acquire rational values for its sides, the degree of difficulty of the task changes dramatically. In this paper a basis is developed, to produce right Pythagorean triangles with rational sides and integral area in a straightforward manner. Determining whether or not a given natural number n is congruent, is equivalent to a search through an ordered 2D array.

scholarly journals On Congruent Numbers

2021 ◽  
Author(s):  
Alex Nguhi
Keyword(s):  
Pythagorean Triples ◽  
Congruent Number ◽  
Number Problem ◽  
Congruent Number Problem

With respect to some classification of Pythagorean triples, if anumber 𝑘 is congruent then it can easily be proven. This expandsthe quest to resolve the congruent number problem. A proposi-tion is put forward on rational sides forming a congruent number.

scholarly journals A GENERALIZATION OF THE CONGRUENT NUMBER PROBLEM

2011 ◽  
Vol 07 (08) ◽  
pp. 2237-2247 ◽  
Author(s):  
LARRY ROLEN
Keyword(s):  
Elliptic Curve ◽  
Related Problem ◽  
Fixed Angle ◽  
Root Numbers ◽  
Tate Conjecture ◽  
Congruent Number ◽  
Number Problem ◽  
Parity Conjecture ◽  
Congruent Number Problem

We study a certain generalization of the classical Congruent Number Problem. Specifically, we study integer areas of rational triangles with an arbitrary fixed angle θ. These numbers are called θ-congruent. We give an elliptic curve criterion for determining whether a given integer n is θ-congruent. We then consider the "density" of integers n which are θ-congruent, as well as the related problem giving the "density" of angles θ for which a fixed n is congruent. Assuming the Shafarevich–Tate conjecture, we prove that both proportions are at least 50% in the limit. To obtain our result we use the recently proven p-parity conjecture due to Monsky and the Dokchitsers as well as a theorem of Helfgott on average root numbers in algebraic families.

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).

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