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One way to prove that there is an infinity of consecutive powerful integers is to just note that there are infinite solutions to the equation $x^2-8y^2=1$, because clearly $x^2$ and $8y^2$ are clearly powerful. But this uses a bit of theory.

I found this problem under an induction worksheet, and all of the surrounding problems are easy. Is there really an elementary solution to the problem?

All of my attempts have been fruitless.

Note: a number is powerful if whenever a prime $p$ divides it we have that $p^2$ also divides it.

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    This is a particularly nasty one. The Pell's equation method seems a lot more natural to me. – 2017-01-11

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Ugh I found it.

The idea is to find an expression such that such that both the experession and the expression plus 1 can be put in terms of powerful stuff.

I found that $4n^2+4n$ is such an expression.

$4n(n+1)$ is powerful if $n$ and $n+1$ are powerful, and $4n^2+4n+1$ is a square.