  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, $(x, y, z) = (k-1, 1, 2).$
  Using the structure of class groups of binary quadratic forms, we prove the
conjecture when $4\Vert k$, with $2k-1$ a prime power and $4\le k\le 1000$.
