Problem 942

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Statement. Let $h(n)$ count the number of powerful (if $p\mid m$ then $p^2\mid m$) integers in $[n^2,(n+1)^2)$. Estimate $h(n)$. In particular is there some constant $c>0$ such that

$$h(n) < (\log n)^{c+o(1)}$$

and, for infinitely many $n$,

$$h(n) >(\log n)^{c-o(1)}?$$

Status. Open.

Source. erdosproblems.com/942, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #942, https://www.erdosproblems.com/942.

References.

• [DLS05] De Koninck, Jean-Marie and Luca, Florian and Shparlinski, Igor E., Powerful numbers in short intervals. Bull. Austral. Math. Soc. (2005), 11-16.
• [DeLu04] De Koninck, Jean-Marie and Luca, Florian, Sur la proximité des nombres puissants. Acta Arith. (2004), 149-157.

Formalization. Statement in formal-conjectures.

Current assessment

The question is open. The results the site credits bound $h(n)$ from below only for infinitely many $n$: Erdős observed that $\limsup h(n)=\infty$, and van Doorn gave a proof in the comments; De Koninck and Luca [DeLu04] showed $h(n)\gg(\log n/\log\log n)^{1/3}$ for infinitely many $n$ and computed the density, about $0.275$, of the $n$ with $h(n)=1$; and Hughes (with AI assistance) observed that their argument, optimized, gives $h(n)\gg\log n/(\log\log n\log\log\log n)$ for infinitely many $n$. None of these gives an upper bound $h(n)<(\log n)^{c+o(1)}$ for all $n$, so none settles the question or an instance of it, and no claim page is owed.

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.

• dekoninck_2004_sur_la_proximite_des_nombres
• dekoninck_2005_powerful_numbers_short_intervals
• dekoninck_2005_powerful_numbers_short_intervals / theorem_1
• dekoninck_2005_powerful_numbers_short_intervals / theorem_2
• dekoninck_2005_powerful_numbers_short_intervals / theorem_3