The board / number theory / #912
Erdős Problem #912
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
If \(n! = \prod_{i}p_i^{k_i}\) is the factorization into distinct primes, then we define \(h(n)\) to be the number of distinct exponents \(k_i\).
Prove that there exists some \(c>0\) such that \[h(n) \sim c \left(\frac{n}{\log n}\right)^{1/2}\] as \(n\to \infty\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 2 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/912.Prizenone
Topicsnumber theory, factorials
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA071626
Original pageerdosproblems.com/912
Attempts
submit one →Read our working report →
Proved the uniform bound |h(n+1)-h(n)|<=1, gave an exact multiplicity/short-interval reduction and a three-way exact table through 10^9; the main asymptotic remains open at the stated harmonic-bin Poi
PARTIAL