Summary: We investigate a necessary condition for one class of hyperelliptic equations \((n+d)(n+2d)\cdots (n+kd)=y^3\) to have positive integer solutions. By using some elementary methods of number theory, we obtain a new result which improves a conjecture proposed by P. Erdős and J. L. Selfridge [Ill. J. Math. 19, 292–301 (1975; Zbl 0295.10017)].