On the Lebesgue-Ramanujan-Nagell type equation $x^2+17^k41^\ell 59^m =2^δy^n$

Abstract: We consider the Diophantine equation $x^2+17^k41^\ell 59^m =2^\delta y^n$ in unknown integer $x\geq 1, y>1, k, \ell, m, \delta\geq 0$ and $n\geq 3$ with $\gcd(x,y)=1$, and we find all its solutions. We use the prominent result of Bilu, Hanrot and Voutier on existence of primitive divisors in Lehmer sequences in combination with elementary number theoretic argument and computer search.