Direct parametrization of Pythagorean triples

Sungkon Chang
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 25, 2019, Number 3, Pages 21—35
DOI: 10.7546/nntdm.2019.25.3.21-35
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Authors and affiliations

Sungkon Chang
Department of Mathematics, Georgia Southern University, Armstrong Campus
11935 Abercorn St, Savannah GA, U.S.A.


If the two axes of symmetry of a quadratic form in two variables have integer coefficients, the reflection across the axes defines a group action on the primitive solutions of the Diophantine equation defined by the quadratic form. In this paper, we introduce quadratic forms with rational axes of symmetry that admit a single set of polynomials which parametrize their primitive solutions up to the reflections.


  • Parametrization of primitive solutions

2010 Mathematics Subject Classification

  • 11D09


  1. Cox, D. A. (2013). Primes of the form x2 + ny2, Wiley.
  2. Cuoco, A. (2000). Meta Problems in Mathematics, College Math. J., 31, 373–378.
  3. Davenport, H. (2000). Multiplicative Number Theory, Springer-Verlag, New York.
  4. Dickson, L. E. (1904). A new extension of Dirichlet’s theorem on prime numbers, Messenger of Math., 33, 155–161.
  5. Frisch, S., & Vaserstein, L.(2008) Parametrization of Pythagorean triples by a single triple of polynomials, J. Pure Appl. Algebra, 212 (1), 271–274.
  6. Frisch, S.,& Lettl, G. (2008). Polynomial parametrization of the solutions of Diophantine equations of genus 0, Funct. Approx. Comment. Math., 39 (2) (Narkiewicz Volume), 205–209.
  7. Gilder, J. (1982). Integer-sided triangles with a 60° angle, Math Gazette, 66, 261–266.
  8. Green, B. & Tao, T. (2010). Linear equations in primes, Ann. Math., 171, 1753–1850.
  9. Jones, G. & Jones, J. M. (1998). Elementary Number Theory, Springer.
  10. Petulante, N., & Kaja, I. (2000). How to generate all integral triangles containing a given angle, Internat. J. Math. & Math. Sci., 24, 569–572.
  11. Read, E. (2006). On integer-sided triangles containing angles of 120° or 60°, Math. Gazette, 90, 299–305.
  12. Selkirk, K. (1983). Integer-sided triangles with angle of 120°, Math. Gazette, 67, 251–255.
  13. Shafarevich, I. R. (1994). Basic Algebraic Geometry 1: Varieties in Projective Space, Springer-Verlag.
  14. Sierpinski, W. (2011) Pythagorean Triangles, Dover Publications.
  15. Silverman, J. H. & Tate, J., (1992). Rational Points on Elliptic Curves, Springer.
  16. Stewart, B. M. (1964). The Theory of Numbers, MacMillan, New York, NY.
  17. Vaserstein, L. (2010). Polynomial parametrization for the solutions of Diophantine equations and arithmetic groups, Ann. of Math., 171, 979–1009.

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Cite this paper

Chang, S. (2019). Direct parametrization of Pythagorean triples. Notes on Number Theory and Discrete Mathematics, 25(3), 21-35, doi: 10.7546/nntdm.2019.25.3.21-35.

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