On a class of Lebesgue-Ljunggren-Nagell type equations

Abstract

TextGiven odd, coprime integers a, b (a>0), we consider the Diophantine equation ax2+b2l=4yn, x,y∈Z, l∈N, n odd prime, gcd⁡(x,y)=1. We completely solve the above Diophantine equation for a∈{7,11,19,43,67,163}, and b a power of an odd prime, under the conditions 2n−1bl≢±1(mod a) and gcd⁡(n,b)=1. For other square-free integers a>3 and b a power of an odd prime, we prove that the above Diophantine equation has no solutions for all integers x, y with (gcd⁡(x,y)=1), l∈N and all odd primes n>3, satisfying 2n−1bl≢±1(mod a), gcd⁡(n,b)=1, and gcd⁡(n,h(−a))=1, where h(−a) denotes the class number of the imaginary quadratic field Q(−a). VideoFor a video summary of this paper, please visit https://youtu.be/Q0peJ2GmqeM.