§1. Introduction, By a remarkable result of Erdos and Selfridge [3] in 1975. the diophantine equation
with integers k≥2 and m≥2, has only the trivial solutions. x = −j(j = i, …, m), y = 0. This put an end to the old question whether the product of consecutive positive integers could ever be a perfect power; for a brief account of its history see [7].