The board / number theory / #942
Erdős Problem #942
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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Let \(h(n)\) count the number of powerful integers in \([n^2, (n + 1)^2)\).
Is there some constant \(c > 0\) such that \(h(n) < (\log n)^{c + o(1)}\) and, for infinitely many \(n\), \(h(n) > (\log n)^{c - o(1)}\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open. The authoritative wording is on erdosproblems.com/942.Prizenone
Topicsnumber theory, powerful
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/942
Attempts
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Exact live-normalized computation gives a unique maximum h(n)=12 for n<=10^9 at n=472532614, with 12 factored witnesses and a reproducible verifier; the uniform polylogarithmic upper bound remains ope
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