Concerning the Solutions of Some Lebesgue-Ramanujan-Nagell Equations

In this paper, we provide a complete classification of the integer solutions to the Diophantine equation $x^2 + 3^a 19^b 73^c = \lambda y^n,$ where \( \lambda = 2^{\delta} \), \( x, y \geq 1 \), \( a, b, c, \delta \geq 0 \), \( n \geq 3 \), \( \gcd(x, y) = 1 \). Our approach combines the Primitive Divisor Theorem for Lehmer sequences, proved by Bilu, Hanrot, and Voutier, with fundamental properties of algebraic integer rings. By employing these methods, we determine all possible solutions in non-negative integers.