Lebesgue–Nagell–Ramanujan conjecture for 274837012705

Let $x,y$ be positive integers and let $n\geq 3$. Consider the Diophantine equation

$$x^2=y^n+274837012705.$$

Conjecture for $274837012705$. Every solution $(x,y,n)$ has one of the following forms:

$$\begin{array}{lll} n=3, & & (x,y)=(524303, 384),\;(526321, 1296),\;(1525921777, 1325424),\\ n=4, & & (x,y)=(526321,216), \\ n=6, & & (x,y)=(526321,36),\\ n=12,& & (x,y)=(526321,6),\\ n=13,& & (x,y)=(536561,6). \end{array}$$

This is one of a series of conjectures concerning solutions of Lebesgue–Nagell–Ramanujan type equations. The surrounding computations determine the integral points on several associated quartic curves, but no resolution is supplied here for the full displayed assertion.