On θ-congruent numbers on real quadratic number fields

Janfada, Ali S.; Salami, Sajad

doi:10.2996/kmj/1436403896

Cited by 4 publications

(3 citation statements)

References 10 publications

(6 reference statements)

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“…For the θ -congruent number elliptic curve E θ N , we denote by T θ N (Q) the torsion subgroup and by r θ N (Q) the Mordell-Weil rank of E θ N (Q). In the recent decades, the notion of θ -congruent numbers has attracted the interests of some mathematicians, consult Dujella et al (2014), Fujiwara (1997Fujiwara ( , 2002, Janfada and Salami (2015), Kan (2000). Among other results, Fujiwara (1997) showed the following relation between θ -congruent numbers and r θ N (Q).…”

Section: Elliptic Curves and â-Congruent Numbersmentioning

confidence: 99%

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

Salami

Zargar

2020

Bull Braz Math Soc, New Series

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A positive integer N is called a θ-congruent number if there is a θ-triangle (a, b, c) with rational sides for which the angle between a and b is equal to θ and its area is N √ r 2 − s 2 , where θ ∈ (0, π), cos(θ) = s/r , and 0 ≤ |s| < r are coprime integers. It is attributed to Fujiwara (Number Theory, de Gruyter, pp 235-241, 1997) that N is a θ-congruent number if and only if the elliptic curve E θ N : y 2 = x(x + (r + s)N)(x − (r − s)N) has a point of order greater than 2 in its group of rational points. Moreover, a natural number N = 1, 2, 3, 6 is a θ-congruent number if and only if rank of E θ N (Q) is greater than zero. In this paper, we answer positively to a question concerning with the existence of methods to create new rational θ-triangle for a θ-congruent number N from given ones by generalizing the Fermat's algorithm, which produces new rational right triangles for congruent numbers from a given one, for any angle θ satisfying the above conditions. We show that this generalization is analogous to the duplication formula in E θ N (Q). Then, based on the addition of two distinct points in E θ N (Q), we provide a way to find new rational θ-triangles for the θ-congruent number N using given two distinct ones. Finally, we give an alternative proof for the Fujiwara's Theorem 2.2 and one side of Theorem 2.3. In particular, we provide a list of all torsion points in E θ N (Q) with corresponding rational θ-triangles.

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Section: Elliptic Curves and â-Congruent Numbersmentioning

confidence: 99%

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

Salami

Zargar

2020

Bull Braz Math Soc, New Series

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“…The essential argument for the proof of the lemma above is contained in Tada [18, Theorem 1] who considered the case θ = π/2 for real quadratic fields K. The analogue for real quadratic fields for any θ with rational cosine in [8] adopts the same approach as in [18]. In the case of real multi-quadratic fields, the proof similarly follows from the following well-known result on elliptic curves.…”

Section: Lemma 23 For Every Subfield K Of R a Natural Number N Is mentioning

confidence: 99%

On -Congruent Numbers Over Real Number Fields

Das

Saikia²

2020

Bull. Aust. Math. Soc.

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The notion of $\theta $ -congruent numbers is a generalisation of congruent numbers where one considers triangles with an angle $\theta $ such that $\cos \theta $ is a rational number. In this paper we discuss a criterion for a natural number to be $\theta $ -congruent over certain real number fields.

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“…For the θ-congruent number elliptic curve E θ N , we denote by T θ N (Q) the torsion subgroup and by r θ N (Q) the Mordell-Weil rank of E θ N (Q). In the recent decades, the notion of θ-congruent numbers has attracted the interests of some mathematicians, consult [2,4,5,6,9]. Among other results, Fujiwara [4] showed the following relation between θ-congruent numbers and r θ N (Q).…”

Section: Elliptic Curves and θ-Congruent Numbersmentioning

confidence: 99%

The $\thera$-congruent numbers elliptic curves via a Fermat-type theorem

Salami,

Zargar

2020

Preprint

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A positive integer N is called a θ-congruent number if there is a θ-triangle (a, b, c) with rational sides for which the angle between a and b is equal to θ and its area is N √ r 2 − s 2 , where θ ∈ (0, π), cos(θ) = s/r, and 0 ≤ |s| < r are coprime integers. It is attributed to Fujiwara [4] that N is a θ-congruent number if and only if the elliptic curve) has a point of order greater than 2 in its group of rational points. Moreover, a natural number N = 1, 2, 3, 6 is a θ-congruent number if and only if rank of E θ N (Q) is greater than zero. In this paper, we answer positively to a question concerning with the existence of methods to create new rational θ-triangle for a θ-congruent number N from given ones by generalizing the Fermat's algorithm, which produces new rational right triangles for congruent numbers from a given one, for any angle θ satisfying the above conditions. We show that this generalization is analogous to the duplication formula in E θ N (Q). Then, based on the addition of two distinct points in E θ N (Q), we provide a way to find new rational θ-triangles for the θ-congruent number N using given two distinct ones. Finally, we give an alternative proof for the Fujiwara's theorem 2.2 and one side of Theorem 2.3. In particular, we provide a list of all torsion points in E θ N (Q) with corresponding rational θ-triangles.

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On θ-congruent numbers on real quadratic number fields

Cited by 4 publications

References 10 publications

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

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

On -Congruent Numbers Over Real Number Fields

The $\thera$-congruent numbers elliptic curves via a Fermat-type theorem

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