An elementary note on three consecutive powerful numbers

Abstract: In the recent work by T. H. Chan, it was shown that there are no three consecutive powerful numbers with the middle term a perfect cube and each of the other two having only a single prime factor raised to an odd power. They get it with Pell equations, elliptic curves, and second-order recurrences. In this paper, with results given by Nagell-Ljunggren, Lebesgue, Ko and some elementary methods, we extend the result to similar cases, whose middle term can be perfect powers for infinitely many powers. With this conclusion, we discuss the solutions to some Diophantine equations.