θ-congruent numbers

Fujiwara, Masahiko

doi:10.1515/9783110809794.235

Cited by 17 publications

(24 citation statements)

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“…Moreover, for any of these cases it is easy to construct series of examples such that the j-invariants of the corresponding elliptic curves are pairwise different. Furthermore, with these examples we answer a question left open in [12] (Remark 1 after Proposition 4). Namely, in [12] it is shown that if n ∈ {1, 2, 3, 6} then n is a θ-congruent number if and only if the rank of the associated elliptic curve is positive; i.e., it is not possible to obtain a corresponding θ-triangle from torsion points on this elliptic curve.…”

Section: Interpretation and Conclusionmentioning

confidence: 77%

“…It also becomes clear from our calculations that two rational points of order > 2 on the curve yield the same triangle if and only if they differ by a 2-torsion element in the Mordell-Weil group of the curve. Moreover, our calculations clarify Theorem 1 and Proposition 4 in [12]. More precisely, we will exactly determine those numbers n occurring as θ-congruent numbers corresponding to torsion elements of the associated elliptic curve, which was left open in [12] and was clarified in the paper [22], of which we were made aware only after finishing our paper.…”

Section: Connections To Elliptic Curvesmentioning

confidence: 83%

“…[11]) and even more generally to that of a θ-congruent number (cf. [12][13][14][15][16]); for an overview see [11]. Even though the concept of a t-congruent number is more natural from a geometric point of view (arising from the search for triangles with rational sides and rational area), the more general concept of a θ-congruent number is more relevant for the purposes of this paper.…”

Section: Definitionmentioning

confidence: 99%

“…The use of a true isomorphism seems more natural and also sheds some new light on the theory of θ-congruent triangles; for example, two rational points of order >2 on the elliptic curve in question give rise to the same θ-congruent triangle if and only if they differ by a 2-torsion element, and a θ-congruent triangle is isosceles if and only if the associated rational point on the curve has order 4 (see Theorem 8 below). Our approach allows us to clarify two statements in [12] and to answer a question left open in that paper, and we also see that the case n = 1 needs to be ruled out in Proposition 5.4 in [11]. (See below for details.…”

Section: Definitionmentioning

confidence: 99%

“…While a thorough analysis of the mapping σ used in [3] together with the two-descent on the elliptic curve E(m, n) may reveal all the interesting phenomena concerning solutions to the concordant form problem from the study of rational points on E(m, n) via σ, the approach via the isomorphism ϕ seems to be much more direct and natural. Also, the correspondence between solutions (r, s, k) of the congruent number problem and rational points of order >2 on the elliptic curve y 2 = x x − (s−r)k x + (s+r)k becomes, via the correspondence with rational squares in arithmetic progressions, more lucid than in [12][13][14]. It also becomes clear from our calculations that two rational points of order > 2 on the curve yield the same triangle if and only if they differ by a 2-torsion element in the Mordell-Weil group of the curve.…”

Section: Connections To Elliptic Curvesmentioning

confidence: 99%

See 4 more Smart Citations

On θ-Congruent Numbers, Rational Squares in Arithmetic Progressions, Concordant Forms and Elliptic Curves

Selder

Spindler

2015

Mathematics

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Abstract:The correspondence between right triangles with rational sides, triplets of rational squares in arithmetic succession and integral solutions of certain quadratic forms is well-known. We show how this correspondence can be extended to the generalized notions of rational θ-triangles, rational squares occurring in arithmetic progressions and concordant forms. In our approach we establish one-to-one mappings to rational points on certain elliptic curves and examine in detail the role of solutions of the θ-congruent number problem and the concordant form problem associated with nontrivial torsion points on the corresponding elliptic curves. This approach allows us to combine and extend some disjoint results obtained by a number of authors, to clarify some statements in the literature and to answer some hitherto open questions.

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Section: Interpretation and Conclusionmentioning

confidence: 77%

Section: Connections To Elliptic Curvesmentioning

confidence: 83%

Section: Definitionmentioning

confidence: 99%

Section: Definitionmentioning

confidence: 99%

Section: Connections To Elliptic Curvesmentioning

confidence: 99%

See 3 more Smart Citations

On θ-Congruent Numbers, Rational Squares in Arithmetic Progressions, Concordant Forms and Elliptic Curves

Selder

Spindler

2015

Mathematics

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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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Four rational squares in arithmetic progressions and a family of elliptic curves with positive Mordell-Weil rank

2020

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We study the question at which relative distances four squares of rational numbers can occur as terms in an arithmetic progression. This number-theoretical problem is seen to be equivalent to finding rational points on certain elliptic curves. Both number-theoretical results and results concerning the associated elliptic curves are derived; i.e., the correspondence between rational squares in arithmetic progressions and elliptic curves is exploited both ways.

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θ-congruent numbers

Cited by 17 publications

References 0 publications

On θ-Congruent Numbers, Rational Squares in Arithmetic Progressions, Concordant Forms and Elliptic Curves

On θ-Congruent Numbers, Rational Squares in Arithmetic Progressions, Concordant Forms and Elliptic Curves

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

Four rational squares in arithmetic progressions and a family of elliptic curves with positive Mordell-Weil rank

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