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The Farey Sum of Pythagorean and Eisenstein Triples

Year 2024, Volume: 12 Issue: 1, 28 - 35, 28.01.2024

Mircea Crasmareanu

https://doi.org/10.36753/mathenot.1316554

Abstract

A composition law, inspired by the Farey addition, is introduced on the set of Pythagorean triples. We study some of its properties as well as two symmetric matrices naturally associated to a given Pythagorean triple. Several examples are discussed, some of them involving the degenerated Pythagorean triple $(1, 0, 1)$. The case of Eisenstein triples is also presented.

Keywords

Farey sum Pythagorean (Eisenstein) triple Symmetric matrix

References

  • [1] Kramer, Jürg., Pippich, A. M. V.: Snapshots of modern mathematics from Oberwolfach: Special values of zeta functions and areas of triangles. Notices of American Mathematical Society, 63(8), 917-922 (2016).
  • [2] Bonahon, F.: Low-Dimensional Geometry: From Euclidean Surfaces to Hyperbolic Knots. American Mathematical Society; Princeton, NJ: Institute for Advanced Study. 2009.
  • [3] Katok, S., Ugarcovici I.: Symbolic dynamics for the modular surface and beyond. Bulletin of American Mathematical Society, New Ser. 44(1), 87-132 (2007).
  • [4] Hatcher, A.: Topology of Numbers. American Mathematical Society. 2022.
  • [5] Jitman, S., Sangwisut, E.: The group of primitive Pythagorean triples and perplex numbers. Mathematics Magazine. 95(4), 285-293 (2022).
  • [6] Crasmareanu, M.: The diagonalization map as submersion, the cubic equation as immersion and Euclidean polynomials. Mediterranean Journal of Mathematics. 19(2), 65 (2022).
  • [7] Barron, E. N.: Game Theory: An Introduction. 2nd. Revised and Enlarged ed., JohnWiley & Sons. 2013.
  • [8] Crasmareanu, M.: Conics from the Cartan decomposition of SO(2; 1). Mathematics. 11(7), 1580 (2023).
  • [9] Pluta, K., Roussillon, T., Coeurjolly, D., Romon P., Kenmochi, Y., Ostromoukhov V.: Characterization of bijective digitized rotations on the hexagonal grid. Journal of Mathematical Imaging and Vision. 60(5), 707-716 (2018).

Year 2024, Volume: 12 Issue: 1, 28 - 35, 28.01.2024

Mircea Crasmareanu

https://doi.org/10.36753/mathenot.1316554

Abstract

References

  • [1] Kramer, Jürg., Pippich, A. M. V.: Snapshots of modern mathematics from Oberwolfach: Special values of zeta functions and areas of triangles. Notices of American Mathematical Society, 63(8), 917-922 (2016).
  • [2] Bonahon, F.: Low-Dimensional Geometry: From Euclidean Surfaces to Hyperbolic Knots. American Mathematical Society; Princeton, NJ: Institute for Advanced Study. 2009.
  • [3] Katok, S., Ugarcovici I.: Symbolic dynamics for the modular surface and beyond. Bulletin of American Mathematical Society, New Ser. 44(1), 87-132 (2007).
  • [4] Hatcher, A.: Topology of Numbers. American Mathematical Society. 2022.
  • [5] Jitman, S., Sangwisut, E.: The group of primitive Pythagorean triples and perplex numbers. Mathematics Magazine. 95(4), 285-293 (2022).
  • [6] Crasmareanu, M.: The diagonalization map as submersion, the cubic equation as immersion and Euclidean polynomials. Mediterranean Journal of Mathematics. 19(2), 65 (2022).
  • [7] Barron, E. N.: Game Theory: An Introduction. 2nd. Revised and Enlarged ed., JohnWiley & Sons. 2013.
  • [8] Crasmareanu, M.: Conics from the Cartan decomposition of SO(2; 1). Mathematics. 11(7), 1580 (2023).
  • [9] Pluta, K., Roussillon, T., Coeurjolly, D., Romon P., Kenmochi, Y., Ostromoukhov V.: Characterization of bijective digitized rotations on the hexagonal grid. Journal of Mathematical Imaging and Vision. 60(5), 707-716 (2018).
There are 9 citations in total.

Details

Primary Language English
Subjects Dynamical Systems in Applications
Journal Section Articles
Authors

Mircea Crasmareanu 0000-0002-5230-2751

Early Pub Date December 8, 2023
Publication Date January 28, 2024
Submission Date June 19, 2023
Acceptance Date December 5, 2023
Published in Issue Year 2024 Volume: 12 Issue: 1

Cite

APA Crasmareanu, M. (2024). The Farey Sum of Pythagorean and Eisenstein Triples. Mathematical Sciences and Applications E-Notes, 12(1), 28-35. https://doi.org/10.36753/mathenot.1316554
AMA Crasmareanu M. The Farey Sum of Pythagorean and Eisenstein Triples. Math. Sci. Appl. E-Notes. January 2024;12(1):28-35. doi:10.36753/mathenot.1316554
Chicago Crasmareanu, Mircea. “The Farey Sum of Pythagorean and Eisenstein Triples”. Mathematical Sciences and Applications E-Notes 12, no. 1 (January 2024): 28-35. https://doi.org/10.36753/mathenot.1316554.
EndNote Crasmareanu M (January 1, 2024) The Farey Sum of Pythagorean and Eisenstein Triples. Mathematical Sciences and Applications E-Notes 12 1 28–35.
IEEE M. Crasmareanu, “The Farey Sum of Pythagorean and Eisenstein Triples”, Math. Sci. Appl. E-Notes, vol. 12, no. 1, pp. 28–35, 2024, doi: 10.36753/mathenot.1316554.
ISNAD Crasmareanu, Mircea. “The Farey Sum of Pythagorean and Eisenstein Triples”. Mathematical Sciences and Applications E-Notes 12/1 (January 2024), 28-35. https://doi.org/10.36753/mathenot.1316554.
JAMA Crasmareanu M. The Farey Sum of Pythagorean and Eisenstein Triples. Math. Sci. Appl. E-Notes. 2024;12:28–35.
MLA Crasmareanu, Mircea. “The Farey Sum of Pythagorean and Eisenstein Triples”. Mathematical Sciences and Applications E-Notes, vol. 12, no. 1, 2024, pp. 28-35, doi:10.36753/mathenot.1316554.
Vancouver Crasmareanu M. The Farey Sum of Pythagorean and Eisenstein Triples. Math. Sci. Appl. E-Notes. 2024;12(1):28-35.
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