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

Abstract

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 $4\le k\le 1000$.