The board / number theory / #419
Erdős Problem #419
SOLVED LEANdatabase updated 2026-01-31
Resolved in some way other than a straight proof or disproof — usual for open-ended questions, or for multi-part problems whose parts went different ways. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
The ratio \(\sigma_0((n+1)!)/\sigma_0(n!)\), where \(\sigma_0\) is the divisor-counting function.
The set \(\{1\} \cup \{1+1/k : k \geq 1\}\).
If \(\tau(n)\) counts the number of divisors of \(n\), then what is the set of limit points of \[ \frac{\tau((n+1)!)}{\tau(n!)}? \] The limit points are exactly \(\{1\} \cup \{1+1/k : k \geq 1\}\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved. The authoritative wording is on erdosproblems.com/419.Prizenone
Topicsnumber theory, factorials
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/419
Attempts
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