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

**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*(*k*^{6n+3}) : *y*^{2} = *x*^{3} + *k*^{6n+3} for some integers *k* and *n* ≥ 0 and prove that their rank is zero and the torsion part is isomorphic to ℤ_{2}. 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

- 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.
- 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.
- Baker, A. (1968). Contributions to the theory of Diophantine equations, II. The Diophantine equation
*y*^{2}=*x*^{3}+*k*. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 263, 193–208. - Brunner, O. (1933). Losungseigenschaften der kubischen diophantischen Gleichung
*Z*^{3}−*y*^{2}=*D*. Inauguraldissertation, Zurich. - Cassels, J. W. S. (1950). The rational solutions of the Diophantine equation
*y*^{2}=*x*^{3}−*d*. Acta Mathematica, 82, 243–273. - Ellison, W. J., Ellison, F., Pesek, J., Stahl, C. E., & Stall, D. S. (1972). The Diophantine equation
*y*^{2}+*k*=*x*^{3}. Journal of Number Theory, 4, 107–117. - Finkelstein, R., & London, H. (1970). On Mordell’s equation
*y*^{2}−*k*=*x*^{3}: An interesting case of Sierpinski. Journal of Number Theory, 2, 310–321. - Fueter, R. (1930). Uber kubische diophantische Gleichungen. Commentarii Mathematici Helvetici, 2, 69–89.
- Ljunggren, W. (1961). The Diophantine equation
*y*^{2}=*x*^{3}−*k*. Acta Arithmetica, 8, 451–465. - Mordell, L. J. (1969). On some Diophantine equations
*y*^{2}=*x*^{3}+*k*with no rational solutions (II). In: Number Theory and Analysis, Springer, Boston, MA, pp. 224–232. - Scholz, A. (1932). Über die Beziehung der Klassenzahl enquadratischer Körper zueinander. Journal für die reine und angewandte Mathematik, 166, 201–203.
- Silverman, J. H. (1986). The Arithmetic of Elliptic Curves. Springer-Verlag.
- Silverman, J. H., & Tate, J .T. (1992). Rational Points on Elliptic Curves, Undergraduate Texts in Mathematics, Springer-Verlag, New York.
- Wu, X. & Qin, Y. (2018). Rational points of Elliptic Curve
*y*^{2}=*x*^{3}+*k*^{3}. Algebra Colloquium, 25, 133–138.

## Related papers

## Cite this paper

Sharma, R., & Bhatter, S. (2021). On the rational solutions of *y*^{2} =*x*^{3} + *k*^{6n+3}. *Notes on Number Theory and Discrete Mathematics*, 27(3), 130-142, DOI: 10.7546/nntdm.2021.27.3.130-142.