Problem 936

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claims/: The 2 claim pages of Problem 936, one per claimant's result; the problem's standing derives from them.

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Statement. Are

$$2^n\pm 1$$

and

$$n!\pm 1$$

powerful (i.e. if $p\mid m$ then $p^2\mid m$) for only finitely many $n$?

Status. Open, the site's label (OPEN). No claim settles the question; both claim pages are conditional on the abc conjecture.

Source. erdosproblems.com/936, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #936, https://www.erdosproblems.com/936.

References.

• [Cr20] P. A. CrowdMath, Applications of the abc conjecture to powerful numbers. arXiv:2005.07321 (2020).
• [CuPa16] D. Cushing and J. E. Pascoe, Powerful numbers and the ABC-conjecture. arXiv:1611.01192 (2016).

Formalization. Statement in formal-conjectures.

Current assessment

Both questions are open unconditionally. The site credits two results under the abc conjecture, each recorded as a conditional claim that settles no standing. Cushing and Pascoe [CuPa16] prove, assuming abc, that only finitely many powerful numbers lie within a fixed distance of a factorial, which answers the second question (claim page); the half for $n!-k$ is left to the reader as an exercise. CrowdMath [Cr20] proves, assuming abc, that $k^n+r$ is powerful only finitely often for fixed coprime positive $k$ and $r$, which covers $2^n+1$ (claim page); the paper does not treat $2^n-1$, so even conditionally the first question is answered only for $2^n+1$, although the site credits the paper with the whole first question.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• crowdmath_2020_applications_abc_conjecture_powerful_numbers
• cushing_2016_powerful_numbers_abc_conjecture
• cushing_2016_powerful_numbers_abc_conjecture / lemma_4_3
• cushing_2016_powerful_numbers_abc_conjecture / theorem_4_1