1106.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

3 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 1106

References:

erdosproblems.com/1106
[ScWi87] Schinzel, A. and Wirsing, E., Multiplicative properties of the partition function. Proc. Indian Acad. Sci. Math. Sci. (1987), 297--303.

Imports (1)

FormalConjecturesUtil

def Erdos1106.p 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) line 37 source
Source documentation
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).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Fintype.card, Nat.Partition
1 instances, coercions and auxiliary constants left out.
```
def p : ℕ → ℕ := fun n => Fintype.card (Nat.Partition n)
…
```
theorem Erdos1106.erdos_1106.parts.i 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 sorry line 46 source
Source documentation
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.
The answer is yes: Schinzel and Wirsing [ScWi87] proved $F(n) \gg \log n$.

depends on sorry what this means
Statement uses: Iff, True, Filter.Tendsto, Nat, Finset.card, Nat.primeFactors, Finset.prod, Finset.Icc, OfNat.ofNat, Erdos1106.p, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
8 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1106.parts.i :
    answer(True) ↔ Tendsto (fun n => #(∏ i ∈ Icc 1 n, p i).primeFactors) atTop atTop := by
…
```
Mentions: Erdos1106.p
theorem Erdos1106.erdos_1106.parts.ii 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 $n$ sorry line 55 source
Source documentation
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 $n$.

depends on sorry what this means
Statement uses: Iff, True, Filter.Eventually, Nat, GT.gt, Finset.card, Nat.primeFactors, Finset.prod, Finset.Icc, OfNat.ofNat, Erdos1106.p, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1106.parts.ii :
    answer(sorry) ↔ ∀ᶠ n in atTop, #(∏ i ∈ Icc 1 n, p i).primeFactors > n := by
…
```
Mentions: Erdos1106.erdos_1106.parts.i, Erdos1106.p