On the rational solutions of y2 =x3 + k6n+3

Richa Sharma and Sanjay Bhatter
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 3, Pages 130–142
DOI: 10.7546/nntdm.2021.27.3.130-142
Download full paper: PDF, 218 Kb

Details

Authors and affiliations

Richa Sharma
Department of Mathematics, Malaviya National Institute of Technology
Jawahar Lal Nehru Marg, Jhalana Gram, Malviya Nagar, Jaipur, Rajasthan 302017, India

Sanjay Bhatter
Department of Mathematics, Malaviya National Institute of Technology
Jawahar Lal Nehru Marg, Jhalana Gram, Malviya Nagar, Jaipur, Rajasthan 302017, India

Abstract

We consider a family of elliptic curves E(k6n+3) : y2 = x3 + k6n+3 for some integers k and n ≥ 0 and prove that their rank is zero and the torsion part is isomorphic to ℤ2. This is an extension of a recent work of Wu and Qin [14].

Keywords

  • Diophantine equation
  • Lebesgue–Nagell type equation
  • Integer solution
  • Lucas sequences
  • Primitive divisors

2020 Mathematics Subject Classification

  • 111G05
  • 14G05
  • 11R29

References

  1. Ankeny, N. C., Artin, E., & Chowla, S. (1952). The class-number of real quadratic number fields. Annals of Mathematics Second Series, 56(3), 479–493.
  2. Baker, A. (1968). Contributions to the theory of Diophantine equations, I. On the representation of integers by binary quadratic forms. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 263, 173–191.
  3. Baker, A. (1968). Contributions to the theory of Diophantine equations, II. The Diophantine equation y2 = x3 + k. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 263, 193–208.
  4. Brunner, O. (1933). Losungseigenschaften der kubischen diophantischen Gleichung Z3y2 = D. Inauguraldissertation, Zurich.
  5. Cassels, J. W. S. (1950). The rational solutions of the Diophantine equation y2 = x3d. Acta Mathematica, 82, 243–273.
  6. Ellison, W. J., Ellison, F., Pesek, J., Stahl, C. E., & Stall, D. S. (1972). The Diophantine equation y2 + k = x3. Journal of Number Theory, 4, 107–117.
  7. Finkelstein, R., & London, H. (1970). On Mordell’s equation y2k = x3: An interesting case of Sierpinski. Journal of Number Theory, 2, 310–321.
  8. Fueter, R. (1930). Uber kubische diophantische Gleichungen. Commentarii Mathematici Helvetici, 2, 69–89.
  9. Ljunggren, W. (1961). The Diophantine equation y2 = x3 − k. Acta Arithmetica, 8, 451–465.
  10. Mordell, L. J. (1969). On some Diophantine equations y2 = x3 + k with no rational solutions (II). In: Number Theory and Analysis, Springer, Boston, MA, pp. 224–232.
  11. Scholz, A. (1932). Über die Beziehung der Klassenzahl enquadratischer Körper zueinander. Journal für die reine und angewandte Mathematik, 166, 201–203.
  12. Silverman, J. H. (1986). The Arithmetic of Elliptic Curves. Springer-Verlag.
  13. Silverman, J. H., & Tate, J .T. (1992). Rational Points on Elliptic Curves, Undergraduate Texts in Mathematics, Springer-Verlag, New York.
  14. Wu, X. & Qin, Y. (2018). Rational points of Elliptic Curve y2 = x3 + k3. Algebra Colloquium, 25, 133–138.

Related papers

Cite this paper

Sharma, R., & Bhatter, S. (2021). On the rational solutions of y2 =x3 + k6n+3. Notes on Number Theory and Discrete Mathematics, 27(3), 130-142, doi: 10.7546/nntdm.2021.27.3.130-142.

Comments are closed.