Integral solutions in arithmetic progression for y2=x3+k

Integral solutions toy ²=x ³+k, where either thex's or they's, or both, are in arithmetic progression are studied. When both thex's and they's are in arithmetic progression, then this situation is completely solved. One set of solutions where they's formed an arithmetic progression of length 4 had already been constructed. In this paper, we construct infinitely many sets of solutions where there are 4x's in arithmetic progression and we disprove Mohanty's Conjecture 8] by constructing infinitely many sets of solutions where there are 4, 5 and 6y's in arithmetic progression.