1054.lean`

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Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

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

6 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 1054

Reference: erdosproblems.com/1054

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100SolvedSet1.lean

def Erdos1054.f Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$ line 34 source
Source documentation
Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: dite, Exists, And, GE.ge, OfNat.ofNat, Eq, Finset.sum, Finset.Iio, Nat.nth, Membership.mem, Finset, Nat.divisors, Classical.propDecidable, Nat.find, Not
9 instances, coercions and auxiliary constants left out.
```
noncomputable def f (n : ℕ) : ℕ :=
…
```
theorem Erdos1054.erdos_1054.parts.i Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$ sorry line 48 source
Source documentation
Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Is it true that $f(n)=o(n)$?
The strong claim that $f(n)=o(n)$ was disproved by Tao in the comments to 468, in which he proves that the upper density of $\{ n : f(n)\leq \delta n\}$ is $\ll \delta^2$.

L4DEL explanation
Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Is it true that $f(n)=o(n)$? The strong claim that $f(n)=o(n)$ was disproved by Tao in the comments to 468, in which he proves that the upper density of $\{ n : f(n)\leq \delta n\}$ is $\ll \delta^2$.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Iff, False, Asymptotics.IsLittleO, Nat, Real, Filter.atTop, Erdos1054.f
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1054.parts.i : answer(False) ↔ (fun n ↦ (f n : ℝ)) =o[atTop] (fun n ↦ (n : ℝ)) := by
…
```
Mentions: Erdos1054.f
theorem Erdos1054.erdos_1054.parts.ii Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$ sorry line 54 source
Source documentation
Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Is it true that $f(n)=o(n)$ for almost all $n$?

depends on sorry what this means
Statement uses: Iff, True, Exists, Set, Nat, And, Set.HasDensity, OfNat.ofNat, Real, Set.univ, Asymptotics.IsLittleO, Filter.atTop, Membership.mem, Erdos1054.f
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
27 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1054.parts.ii : answer(sorry) ↔ ∃ (A : Set ℕ), A.HasDensity 1 ∧
    (fun (n : A) ↦ (f ↑n : ℝ)) =o[atTop] (fun n ↦ (n : ℝ)) := by
…
```
Mentions: Erdos1054.f
theorem Erdos1054.erdos_1054.parts.iii Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$ sorry line 61 source
Source documentation
Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Is it true that $\limsup f(n)/n=\infty$?

depends on sorry what this means
Statement uses: Iff, True, Exists, Set, Nat, And, Set.HasDensity, OfNat.ofNat, Real, Set.univ, Eq, EReal, Filter.limsup, HDiv.hDiv, Erdos1054.f, Filter.atTop, Top.top
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
37 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1054.parts.iii : answer(sorry) ↔ ∃ (A : Set ℕ), A.HasDensity 1 ∧
    atTop.limsup (fun n ↦ (f n : EReal) / n) = ⊤ := by
…
```
Mentions: Erdos1054.f
theorem Erdos1054.f_undefined_at_2 Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$ line 68 source
Source documentation
Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Show that $f$ is undefined at $n=2$, i.e. we get the junk value $0$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Erdos1054.f, OfNat.ofNat
Proof uses: Eq.mpr, dite, Exists, And, GE.ge, Finset.sum, Finset.Iio, Nat.nth, Membership.mem, Finset, Nat.divisors, Classical.propDecidable, Nat.find, Not, id, congrArg, Erdos1054.f.eq_1, dif_neg, False, Ne and 29 more
37 instances, coercions and auxiliary constants left out.

Tactic trace: 14 steps
Open to load the checked before-and-after goal states.
```
theorem f_undefined_at_2 : f 2 = 0 := by
…
```
Mentions: Erdos1054.f
theorem Erdos1054.f_undefined_at_3 Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$ line 87 source
Source documentation
Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Show that $f$ is undefined at $n=5$, i.e. we get the junk value $0$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Erdos1054.f, OfNat.ofNat
Proof uses: Eq.mpr, dite, Exists, And, GE.ge, Finset.sum, Finset.Iio, Nat.nth, Membership.mem, Finset, Nat.divisors, Classical.propDecidable, Nat.find, Not, id, congrArg, Erdos1054.f.eq_1, dif_neg, False, Ne and 77 more
81 instances, coercions and auxiliary constants left out.

Tactic trace: 98 steps
Open to load the checked before-and-after goal states.
```
theorem f_undefined_at_3 : f 5 = 0 := by
…
```
Mentions: Erdos1054.f