16.lean`

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

6 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 16

References:

erdosproblems.com/16
[Ch23] Chen, Yong-Gao, A conjecture of Erdős on $p+2^k$. arXiv:2312.04120 (2023).
[Er50] Erdős, P., On integers of the form $2^k+p$ and some related problems. Summa Brasil. Math. (1950), 113-123.
[Ro34] Romanoff, N. P., Über einige Sätze der additiven Zahlentheorie. Math. Ann. (1934), 668-678.

Imports (1)

FormalConjecturesUtil

def Erdos16.Erdos16Set The set of odd integers not of the form $2^k+p$ line 41 source
Source documentation
The set of odd integers not of the form $2^k+p$.

standard axioms: propext what this means
Statement uses: Set, Nat
Proof uses: Set.ofPred, And, Odd, Not, Exists, Nat.Prime, Eq, HAdd.hAdd, HPow.hPow, OfNat.ofNat
8 instances, coercions and auxiliary constants left out.
```
def Erdos16Set : Set ℕ :=
…
```
def Erdos16.density_zero A set of natural numbers has density 0 line 47 source
Source documentation
A set of natural numbers has density 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat
Proof uses: Filter.Tendsto, Real, HDiv.hDiv, Nat.count, Membership.mem, Classical.propDecidable, Filter.atTop, nhds, OfNat.ofNat
12 instances, coercions and auxiliary constants left out.
```
def density_zero (S : Set ℕ) : Prop :=
…
```
theorem Erdos16.erdos_16 Is the set of odd integers not of the form $2^k+p$ the union of an infinite arithmetic progression and a set of density $0$? sorry line 62 source
Source documentation
Is the set of odd integers not of the form $2^k+p$ the union of an infinite arithmetic progression and a set of density $0$?
Erdős called this conjecture "rather silly".
Chen [Ch23] has proved the answer is no.
This was formalized in Lean by Chin using Aristotle.

depends on sorry what this means
Statement uses: Iff, False, Exists, Set, Nat, And, Eq, Erdos16.Erdos16Set, Union.union, GT.gt, OfNat.ofNat, Set.ofPred, HAdd.hAdd, HMul.hMul, Erdos16.density_zero
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_16 :
    answer(False) ↔
      ∃ A B : Set ℕ, Erdos16Set = A ∪ B ∧
        (∃ a d : ℕ, d > 0 ∧ A = { x | ∃ m : ℕ, x = a + m * d }) ∧
        density_zero B := by
…
```
Mentions: Erdos16.Erdos16Set, Erdos16.density_zero
def Erdos16.positive_lower_density A set of natural numbers has positive lower density line 72 source
Source documentation
A set of natural numbers has positive lower density.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat
Proof uses: LT.lt, EReal, OfNat.ofNat, Filter.liminf, HDiv.hDiv, Real, Nat.count, Membership.mem, Classical.propDecidable, Filter.atTop
17 instances, coercions and auxiliary constants left out.
```
def positive_lower_density (S : Set ℕ) : Prop :=
…
```
theorem Erdos16.erdos_16.variant.romanoff Romanoff [Ro34] showed that the set of odd integers of this form has positive density sorry line 80 source
Source documentation
Romanoff [Ro34] showed that the set of odd integers of this form has positive density.

depends on sorry what this means
Statement uses: Set, Nat, Set.ofPred, And, Odd, Exists, Nat.Prime, Eq, HAdd.hAdd, HPow.hPow, OfNat.ofNat, Erdos16.positive_lower_density
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_16.variant.romanoff :
    let S := { n : ℕ | Odd n ∧ ∃ k p : ℕ, p.Prime ∧ n = 2 ^ k + p }
…
```
theorem Erdos16.erdos_16.variant.erdos Using covering congruences Erdős [Er50] proved that the set of odd integers which are not of this form contains an infinite arithmetic progression sorry line 90 source
Source documentation
Using covering congruences Erdős [Er50] proved that the set of odd integers which are not of this form contains an infinite arithmetic progression.

depends on sorry what this means
Statement uses: Exists, Nat, And, GT.gt, OfNat.ofNat, LE.le, Set, Set.ofPred, Eq, HAdd.hAdd, HMul.hMul, Erdos16.Erdos16Set
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_16.variant.erdos :
    ∃ a d : ℕ, d > 0 ∧ { x | ∃ m : ℕ, x = a + m * d } ⊆ Erdos16Set := by
…
```
Mentions: Erdos16.Erdos16Set