An Algorithm for Solving a Family of Fourth-Degree Diophantine Equations that Satisfy Runge’s Condition

Abstract

This paper proposes an algorithmic implementation of the elementary version of Runge’s method for a family of fourth-degree Diophantine equations in two unknowns. Any Diophantine equation of the fourth degree the leading homogeneous part of which is decomposed into a product of linear and cubic polynomials can be reduced to equations of the type considered in this paper. The corresponding algorithm (in its optimized version) is implemented in the PARI/GP computer algebra system.

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

REFERENCES

  1. 1

    PARI/GP homepage. https://pari.math.u-bordeaux.fr.

  2. 2

    Mordell, L.J., Diophantine Equations, London: Academic, 1969.

    Google Scholar 

  3. 3

    Sprindzhuk, V.G., Klassicheskie diofantovy uravneniya ot dvukh neizvestnykh (Classical Diophantine Equations in Two Unknowns), Moscow: Nauka, 1982.

  4. 4

    Masser, D.W., Auxiliary Polynomials in Number Theory, Cambridge University Press, 2016.

    Google Scholar 

  5. 5

    Runge, C., Ueber ganzzahlige Lösungen von Gleichungen zwischen zwei Veränderlichen, J. Reine Angew. Math., 1887, vol. 100, pp. 425–435.

    MathSciNet  MATH  Google Scholar 

  6. 6

    Walsh, P.G., A quantitative version of Runge’s theorem on Diophantine equations, Acta Arith., 1992, vol. 62, pp. 157–172.

    MathSciNet  Article  Google Scholar 

  7. 7

    Poulakis, D., A simple method for solving the Diophantine equation \({{Y}^{2}} = {{X}^{4}} + a{{X}^{3}} + b{{X}^{2}} + cX + d\), Elem. Math., 1999, vol. 54, pp. 32–36.

    MathSciNet  Article  Google Scholar 

  8. 8

    Tengely, Sz., On the Diophantine equation \(F(x)\) = G(y), Acta Arith., 2003, vol. 110, pp. 185–200.

    MathSciNet  Article  Google Scholar 

  9. 9

    Osipov, N.N. and Gulnova, B.V., An algorithmic implementation of Runge’s method for cubic Diophantine equations, J. Sib. Fed. Univ. Math. Phys., 2018, vol. 11, pp. 137–147.

    MathSciNet  Article  Google Scholar 

  10. 10

    Beukers, F. and Tengely, Sz., An implementation of Runge’s method for Diophantine equations. https://arxiv.org/abs/math/0512418v1.

  11. 11

    Osipov, N.N., Runge’s method for equations of degree four: An elementary approach, Mat. Prosveshchenie, 2015, vol. 3, no. 19, pp. 178–198.

    Google Scholar 

  12. 12

    Osipov, N.N. and Medvedeva, M.I., An elementary algorithm for solving a Diophantine equation of degree four with Runge’s condition, J. Sib. Fed. Univ. Math. Phys., 2019, vol. 12, pp. 331–341.

    MathSciNet  Google Scholar 

  13. 13

    Osipov, N.N. and Dalinkevich, S.D., An algorithm for solving a quartic Diophantine equation satisfying Runge’s condition, Lect. Notes Comput. Sci., 2019, vol. 11661, pp. 377–392.

    MathSciNet  Article  Google Scholar 

  14. 14

    Stroeker, R.J. and de Weger, B.M.M., Solving elliptic Diophantine equations: The general cubic case, Acta Arith., 1999, vol. 87, pp. 339–365.

    MathSciNet  Article  Google Scholar 

  15. 15

    Masser, D.W., Polynomial bounds for Diophantine equations, Am. Math. Monthly, 1986, vol. 93, pp. 486–488.

    Article  Google Scholar 

  16. 16

    Osipov, N.N., Mechanical proof of planimetric theorems of rational type, Program. Comput. Software, 2014, vol. 40, pp. 71–78.

    MathSciNet  Article  Google Scholar 

  17. 17

    Bryuno, A.D., Stepennaya geometriya v algebraicheskikh i differentsial’nykh uravneniyakh (Power Geometry in Algebraic and Differential Equations), Moscow: Fizmatlit, 1998.

  18. 18

    Bryuno, A.D. and Bakhtin, A.B., Resolution of an algebraic singularity by power geometry algorithms, Program. Comput. Software, 2012, vol. 38, no. 2, pp. 57–72.

    MathSciNet  Article  Google Scholar 

Download references

Funding

This work was supported by the Krasnoyarsk Mathematical Center and financed by the Ministry of Science and Higher Education of the Russian Federation in the framework of the establishment and development of regional Centers for Mathematics Research and Education (agreement no. 075-02-2020-1534/1).

Author information

Affiliations

Authors

Corresponding authors

Correspondence to N. N. Osipov or A. A. Kytmanov.

Additional information

Translated by Yu. Kornienko

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Osipov, N.N., Kytmanov, A.A. An Algorithm for Solving a Family of Fourth-Degree Diophantine Equations that Satisfy Runge’s Condition. Program Comput Soft 47, 29–33 (2021). https://doi.org/10.1134/S0361768821010060

Download citation