A binary quadratic approach to $X^2+(2k-1{)}^Y=k^Z$

A conjecture of N. Terai states that for any integer $k>1$, the equation $x^2+(2k-1{)}^y=k^z$ has only one solution, namely, $(x,y,z)=(k-1,1,2)\mathrm{.}$ Using the structure of class groups of binary quadratic forms, we prove the conjecture when $4\mathrm{\parallel }k$, with $2k-1$ a prime power and $4\le k\le 1000$.