292.lean`

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

6 declarations, 5 contain sorry, 0 declared as axioms

Erdős Problem 292

References:

erdosproblems.com/292
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
[Ma00] Martin, Greg, Denser Egyptian fractions. Acta Arith. (2000), 231-260.

Imports (1)

FormalConjecturesUtil

def Erdos292.A The set $A$ of $n\in \mathbb{N}$ such that there exist $1\leq m_1<\cdots <m_k=n$ with $\sum\tfrac{1}{m_i}=1$ line 38 source
Source documentation
The set $A$ of $n\in \mathbb{N}$ such that there exist $1\leq m_1<\cdots <m_k=n$ with $\sum\tfrac{1}{m_i}=1$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat
Proof uses: Set.ofPred, Exists, Finset, And, LE.le, Finset.Icc, OfNat.ofNat, Membership.mem, Eq, Rat, Finset.sum, HDiv.hDiv
14 instances, coercions and auxiliary constants left out.
```
def A : Set ℕ :=
…
```
theorem Erdos292.erdos_292 sorry line 59 source
depends on sorry what this means
Statement uses: Iff, True, Set.HasDensity, Nat, Erdos292.A, OfNat.ofNat, Real, Set.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
13 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos292.erdos_292.variants.martin Martin [Ma00] proved that if $B=\mathbb{N}\backslash A$ then $\frac{\lvert B\cap [1,x]\rvert}{x}\asymp \frac{\log\log x}{\log x}$ sorry line 65 source
Source documentation
Martin [Ma00] proved that if $B=\mathbb{N}\backslash A$ then $\frac{\lvert B\cap [1,x]\rvert}{x}\asymp \frac{\log\log x}{\log x}$.

depends on sorry what this means
Statement uses: Asymptotics.IsTheta, Nat, Real, Filter.atTop, HDiv.hDiv, Set.ncard, Inter.inter, Set, Compl.compl, Erdos292.A, Set.Icc, OfNat.ofNat, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_292.variants.martin :
    (fun x : ℕ ↦ ((Aᶜ ∩ Set.Icc 1 x).ncard : ℝ) / x) =Θ[atTop]
      fun x ↦ Real.log (Real.log x) / Real.log x := by
…
```
theorem Erdos292.erdos_292.variants.mul Straus observed that $A$ is closed under multiplication sorry line 72 source
Source documentation
Straus observed that $A$ is closed under multiplication.

depends on sorry what this means
Statement uses: Nat, Membership.mem, Set, Erdos292.A, HMul.hMul
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
7 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_292.variants.mul : ∀ m ∈ A, ∀ n ∈ A, m * n ∈ A := by
…
```
Mentions: Erdos292.A
theorem Erdos292.erdos_292.variants.prime_pow $A$ does not contain any prime power sorry line 77 source
Source documentation
$A$ does not contain any prime power.

depends on sorry what this means
Statement uses: Nat, Membership.mem, Set, Erdos292.A, Not, IsPrimePow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_292.variants.prime_pow : ∀ n ∈ A, ¬ IsPrimePow n := by
…
```
Mentions: Erdos292.A
theorem Erdos292.erdos_292.variants.two_mul van Doorn observed that if $n\in A$ (with $n>1$) then $2n\in A$ also sorry line 82 source
Source documentation
van Doorn observed that if $n\in A$ (with $n>1$) then $2n\in A$ also.

depends on sorry what this means
Statement uses: Nat, Membership.mem, Set, Erdos292.A, LT.lt, OfNat.ofNat, HMul.hMul
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_292.variants.two_mul : ∀ n ∈ A, 1 < n → 2 * n ∈ A := by
…
```
Mentions: Erdos292.A