The board / number theory / #400
Erdős Problem #400
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 any \(k\geq 2\) let \(g_k(n)\) denote the maximum value of \((a_1+\cdots+a_k)-n\) where \(a_1,\ldots,a_k\) are integers such that \(a_1!\cdots a_k! \mid n!\).
Can one show that \(\sum_{n\leq x}g_k(n) \sim c_k x\log x\) for some constant \(c_k\)?
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/400.Prizenone
Topicsnumber theory, factorials
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/400
Attempts
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Live page status OPEN; it lists SamKorsky and arkyang as currently working on the problem, so the required no-collision rule applies.
SKIPPED