On the Diophantine equation dx²+p^2aq^2b=4y^p

2021/06/03 by Chakraborty, Kalyan, Hoque, Azizul
#11D41 #11D62 #11Y50 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2106.01964

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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