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On the Diophantine Equation \(dx^2+p^{2a}q^{2b}=4y^p\)

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Abstract

We investigate the solvability of the Diophantine equation in the title, where \(d>1\) is a square-free integer, p, q are distinct odd primes and x, y, a, b are unknown positive integers with \(\gcd (x,y)=1\). We describe all the integer solutions of this equation, and then use the main finding to deduce some results concerning the integers solutions of some of its variants. The methods adopted here are elementary in nature and are primarily based on the existence of the primitive divisors of certain Lehmer numbers.

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Acknowledgements

This work is supported by the grants SERB MATRICS Project No. MTR/2017/001006 and SERB-NPDF (PDF/2017/001958), Govt. of India. The authors are grateful to the anonymous referee for careful reading and valuable comments which have helped to improve this paper.

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  1. Kerala School of Mathematics, Kozhikode, 673571, Kerala, India

    Kalyan Chakraborty

  2. Department of Mathematics, Rangapara College, Rangapara, Sonitpur, 784505, India

    Azizul Hoque

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  1. Kalyan Chakraborty
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  2. Azizul Hoque
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Correspondence to Azizul Hoque.

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Chakraborty, K., Hoque, A. On the Diophantine Equation \(dx^2+p^{2a}q^{2b}=4y^p\). Results Math 77, 18 (2022). https://doi.org/10.1007/s00025-021-01560-w

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