The board / number theory / #18
Erdős Problem #18
practical numbers
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).
For a practical number \(n\), \(h(n)\) is the maximum over all \(1 ≤ m ≤ n\) of the minimum number of divisors of \(n\) needed to represent \(m\) as a sum of distinct divisors.
**Conjecture 1.** Are there infinitely many practical numbers \(m\) such that \(h(m) < (\log \log m)^{O(1)}\)? More precisely: does there exist a constant \(C > 0\) such that for infinitely many practical numbers \(m\), we have \(h(m) < (\log \log m)^C\)?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 4 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/18.Prizenone
Topicsnumber theory, divisors, factorials
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA005153
Original pageerdosproblems.com/18
Attempts
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The live page is OPEN - $250 but lists 1 claimed proof (a partial proof claimed by Liam Price, submitted 2026-07-24, asserting the first question) and Sam_Petkov as currently working on the problem; m
SKIPPED