Erdős Problem #1106 — open problem in number theory

Erdős Problem #1106

Let p(n) be the partition number of n and F(n) be the number of distinct prime factors of ∏_i= 1 ^ n p(n), then F(n) tends to infinity when n tends to infinity.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite

@misc{cairn-erdos-1106,
  title        = {Erdős Problem #1106},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1106}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

erdos_1106.parts.i. 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.

erdos_1106.parts.ii. Let $p (n)$ be the partition number of $n$ and $F (n)$ be the number of distinct prime factors of $\prod_{i = 1}^{n} p (n)$ , $F (n) > n$ for sufficiently large $n$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1106» (2 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_1106.parts.i :
    answer(sorry) ↔ Tendsto (fun n => #(∏ i ∈ Icc 1 n, p i).primeFactors) atTop atTop

theorem erdos_1106.parts.ii :
    answer(sorry) ↔ ∀ᶠ n in atTop, #(∏ i ∈ Icc 1 n, p i).primeFactors > n

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already — check erdosproblems.com/1106. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/1064

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.