A Note on a Lebesgue-Ramanujan-Nagell Type Diophantine Equation

Abstract

We determine all solutions of \[ x^2+3^a7^b37^c=\lambda y^n, \] where $\lambda\in\{1,2,4\}$, $x,y\geq1$, $a,b,c\geq0$, $n\geq3$, and $\gcd(x,y)=1$, subject to the following parity condition: when $\lambda\in\{1,2\}$ and neither $3$ nor $4$ divides $n$, the integer $y$ is assumed to be odd. The cases divisible by $3$ or $4$ are reduced to the determination of $S$-integral points on elliptic and quartic curves, with $S=\{3,7,37\}$. For the remaining exponents, a reduction to odd prime exponents is combined with the primitive divisor theorem for Lehmer sequences and the corrected classification of defective Lehmer pairs. All computationally obtained solutions are then checked directly in the original equation.