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Claim. Assume the abc conjecture. Then for fixed positive integers $k$ and $r$ with $\gcd(k,r)=1$ there are only finitely many powerful numbers of the form $k^n+r$ (Theorem 2.3 of P. A. CrowdMath, Applications of the abc conjecture to powerful numbers, arXiv:2005.07321, posted 2020-05-15), which answers Problem 4 of Cushing and Pascoe, that $2^n+1$ is powerful only finitely often. The paper's Theorem 2.4 shows, under the same hypothesis, that $(n!)^r+k$ is powerful only finitely often for fixed positive $r$ and $k$, which covers $n!+1$. The source card is crowdmath_2020_applications_abc_conjecture_powerful_numbers.

What it leaves open. Theorem 2.3 needs $r$ positive, and the paper does not treat $2^n-1$, although the site credits it with the whole first question of Problem 936, on $2^n\pm1$. Even conditionally, the claim answers only the case $2^n+1$ of that question.

Hypothesis. The abc conjecture: for every $\epsilon>0$ there is a constant $K_\epsilon$ such that coprime positive integers $a+b=c$ satisfy $c<K_\epsilon\operatorname{rad}(abc)^{1+\epsilon}$. It is unproved, so the claim gives no unconditional answer and settles no standing of the problem.

Depends on. No other wiki page; the claim rests on the cited preprint and the hypothesis stated above.

Acceptance. None. The paper is an arXiv preprint with no journal version found, and the site labels the problem OPEN, so the site's commentary crediting the paper is not acceptance.