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

Abstract

A positive integer N is called a \(\theta \)-congruent number if there is a \({\theta }\)-triangle (abc) with rational sides for which the angle between a and b is equal to \(\theta \) and its area is \(N \sqrt{r^2-s^2}\), where \(\theta \in (0, \pi )\), \(\cos (\theta )=s/r\), and \(0 \le |s|<r\) are coprime integers. It is attributed to Fujiwara (Number Theory, de Gruyter, pp 235–241, 1997) that N is a \({\theta }\)-congruent number if and only if the elliptic curve \(E_N^{\theta }: y^2=x (x+(r+s)N)(x-(r-s)N)\) has a point of order greater than 2 in its group of rational points. Moreover, a natural number \(N\ne 1,2,3,6\) is a \({\theta }\)-congruent number if and only if rank of \(E_N^{\theta }({{\mathbb {Q}}})\) is greater than zero. In this paper, we answer positively to a question concerning with the existence of methods to create new rational \({\theta }\)-triangle for a \({\theta }\)-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 \({\theta }\) satisfying the above conditions. We show that this generalization is analogous to the duplication formula in \(E_N^{\theta }({{\mathbb {Q}}})\). Then, based on the addition of two distinct points in \(E_N^{\theta }({{\mathbb {Q}}})\), we provide a way to find new rational \({\theta }\)-triangles for the \({\theta }\)-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 \(E_N^{\theta }({{\mathbb {Q}}})\) with corresponding rational \({\theta }\)-triangles.

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Acknowledgements

The first named author would likes to thank for the hospitality of Institute of Advanced Studies in Basic Sciences (IASBS) during his sabbatical year as a postdoctoral researcher supported by the Iranian National Elites Foundation. The second named author also thanks for the hospitality and partial financial support of IASBS. Both authors would express their gratitude for the referee’s valuable comments which improved the quality of the work.

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Correspondence to Sajad Salami.

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Salami, S., Zargar, A.S. The \({\theta }\)-Congruent Number Elliptic Curves via Fermat-type Algorithms. Bull Braz Math Soc, New Series (2020). https://doi.org/10.1007/s00574-020-00237-2

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Keywords

  • \(\theta \)-Congruent number
  • Rational \({\theta }\)-triangle
  • Elliptic curve
  • Fermat’s algorithm

Mathematics Subject Classification

  • 11G05
  • 14H52