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


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.

This is a preview of subscription content, access via your institution.


  1. Chan, S.: Rational right triangles of a given area. Am. Math. Monthly 125, 689–703 (2018)

    MathSciNet  Article  Google Scholar 

  2. Dujella, A., Janfada, A.S., Peral, C.J., Salami, S.: On the high rank \(\pi /3\) and \(2\pi /3\)-congruent number elliptic curves. Rock. Mt. J. Math. 44, 1867–1880 (2014)

    MathSciNet  Article  Google Scholar 

  3. Fermat, P.: Fermat’s Diophanti Alex. Arith., 1670 in Oeuvres III, (Ministère de l’instruction publique, ed.), Gauthier-Villars et fils, Paris 254–256 (1896)

  4. Fujiwara, M.: \({{\theta }} \)-congruent numbers, in: Number Theory, K. Győry, A. Pethő and V. Sós (eds.), de Gruyter 235–241 (1997)

  5. Fujiwara, M.: Some properties of \({{\theta }} \)-congruent numbers. Nat. Sci. Rep. Ochanomizu Univ. 52, 1–8 (2002)

    MathSciNet  Google Scholar 

  6. Halbeisen, L., Hungerbühler, N.: A theorem of Fermat and congruent number curves. Hardy-Ramanujan J. 41, 15–21 (2018)

    MathSciNet  MATH  Google Scholar 

  7. Hungerbühler, N.: Proof of a conjecture of Lewis Carroll. Math. Magn. 69, 182–184 (1996)

    MathSciNet  Article  Google Scholar 

  8. Janfada, A.S., Salami, S.: On \({{\theta }} \)-congruent numbers on real quadratic number fields. Kodai Math. J. 38, 352–364 (2015)

    MathSciNet  Article  Google Scholar 

  9. Kan, M.: \({{\theta }} \)-congruent numbers and elliptic curves. Acta Arith. 94, 153–160 (2000)

    MathSciNet  Article  Google Scholar 

  10. Koblitz, N.: Introduction to Elliptic Curves and Modular Forms. Springer, New York (1993)

    Google Scholar 

  11. Mazur, B.: Modular curves and the Eisenstein ideal. Publ. Math. Inst. Hautes Études Sci. 47, 33–186 (1977)

    MathSciNet  Article  Google Scholar 

  12. Silverman, J.H.: The Arithmetic of Elliptic Curves. Springer, New York (2009)

    Google Scholar 

Download references


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.

Author information



Corresponding author

Correspondence to Sajad Salami.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

About this article

Verify currency and authenticity via CrossMark

Cite this article

Salami, S., Zargar, A.S. The \({\theta }\)-Congruent Number Elliptic Curves via Fermat-type Algorithms. Bull Braz Math Soc, New Series (2020).

Download citation


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

Mathematics Subject Classification

  • 11G05
  • 14H52