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

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## 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

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## 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.