The board / number theory / #1106
Erdős Problem #1106
OPENdatabase updated 2025-11-17
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).
Let \(p(n)\) be the partition number of \(n\) and \(F(n)\) be the number of distinct prime factors of \(\prod_{i= 1} ^ {n} p(n)\), then \(F(n)\) tends to infinity when \(n\) tends to infinity. The answer is yes: Schinzel and Wirsing [ScWi87] proved \(F(n) \gg \log n\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/1106.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/1106
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Complete factorization gives F(115)=115 and F(116)=117. From there, one fresh proven-prime divisor of each partition number p(n) certifies F(n)>=n+1 for every 116<=n<=10,100. The 100 new steps extend the published finite frontier; the eventual inequality remains open.
PARTIAL