FormalConjectures.ErdosProblems.«1106»

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import FormalConjecturesUtil

Erdős Problem 1106

Reference: erdosproblems.com/1064


open Nat Finset Filter Topology


namespace Erdos1106

The partition function p(n) is the number of ways to write n as a sum of positive integers (where the order of the summands does not matter).

def p : ℕ → ℕ := fun n => Fintype.card (Nat.Partition n)

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.

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_1106.parts.i :
    answer(sorry) ↔ Tendsto (fun n => #(∏ i ∈ Icc 1 n, p i).primeFactors) atTop atTop := by⊢ True ↔ Tendsto (fun n => #(∏ i ∈ Icc 1 n, p i).primeFactors) atTop atTop
  sorryAll goals completed! 🐙

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

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_1106.parts.ii :
    answer(sorry) ↔ ∀ᶠ n in atTop, #(∏ i ∈ Icc 1 n, p i).primeFactors > n := by⊢ True ↔ ∀ᶠ (n : ℕ) in atTop, #(∏ i ∈ Icc 1 n, p i).primeFactors > n
  sorryAll goals completed! 🐙


end Erdos1106