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
Full paper (PDF, 218 Kb)

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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

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