There is a very rich literature on perfect powers or almost perfect powers in products of the form
m(m + d)
are coprime positive integers and k ≥ 3. By a conjecture, such a product is never a perfect
-th power if
> 2. In the classical case
= 1 the conjecture has been proved by Erdős and Selfridge . The general case
≥ 1 seems to be very hard, then there are only partial results; for survey papers on results obtained before 2006 we refer to Tijdeman – , Shorey and Tijdeman [43, 44], Shorey –_and Győry [15, 16].