141.lean`

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

10 declarations, 5 contain sorry, 0 declared as axioms

Erdős Problem 141

References:

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100SolvedSet1.lean

def Erdos141.Set.IsPrimeProgressionOfLength The predicate that a set s consists of l consecutive primes (possibly infinite) line 36 source
Source documentation
The predicate that a set s consists of l consecutive primes (possibly infinite). This predicate does not assert a specific value for the first term.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat, ENat
Proof uses: Exists, And, Eq, ENat.card, Set.ofPred, LT.lt, Nat.nth, Nat.Prime, HAdd.hAdd
6 instances, coercions and auxiliary constants left out.
```
def Set.IsPrimeProgressionOfLength (s : Set ℕ) (l : ℕ∞) : Prop :=
…
```
theorem Erdos141.first_three_odd_primes The first three odd primes are an example of three consecutive primes line 45 source
Source documentation
The first three odd primes are an example of three consecutive primes.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Erdos141.Set.IsPrimeProgressionOfLength, Insert.insert, Nat, Set, OfNat.ofNat, Singleton.singleton, ENat
Proof uses: Exists.intro, And, Eq, ENat.card, Set.ofPred, Exists, LT.lt, Nat.nth, Nat.Prime, HAdd.hAdd, And.intro, of_eq_true, Eq.trans, True, congrFun', congrArg, Fintype.card, ENat.card_eq_coe_fintype_card, Fintype.ofFinset, Finset and 32 more
68 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
```
theorem first_three_odd_primes : ({3, 5, 7} : Set ℕ).IsPrimeProgressionOfLength 3 := by
…
```
def Erdos141.Set.IsAPAndPrimeProgressionOfLength The predicate that a set s is both an arithmetic progression of length l and a progression of l consecutive primes line 56 source
Source documentation
The predicate that a set s is both an arithmetic progression of length l and a progression of l consecutive primes.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat
Proof uses: And, Set.IsAPOfLength, ENat, Erdos141.Set.IsPrimeProgressionOfLength
3 instances, coercions and auxiliary constants left out.
```
def Set.IsAPAndPrimeProgressionOfLength (s : Set ℕ) (l : ℕ) :=
…
```
theorem Erdos141.exists_three_consecutive_primes_in_ap There are 3 consecutive primes in arithmetic progression line 63 source
Source documentation
There are 3 consecutive primes in arithmetic progression.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Exists, Set, Nat, Erdos141.Set.IsAPAndPrimeProgressionOfLength, OfNat.ofNat
Proof uses: Exists.intro, Insert.insert, Singleton.singleton, And.intro, Set.IsAPOfLength, ENat, Erdos141.Set.IsPrimeProgressionOfLength, Set.IsAPOfLengthWith, id, Eq, ENat.card, Set.ofPred, LT.lt, HAdd.hAdd, HSMul.hSMul, of_eq_true, Eq.trans, True, congrFun', congrArg and 38 more
75 instances, coercions and auxiliary constants left out.

Tactic trace: 13 steps
Open to load the checked before-and-after goal states.
```
theorem exists_three_consecutive_primes_in_ap : ∃ (s : Set ℕ), s.IsAPAndPrimeProgressionOfLength 3 := by
…
```
theorem Erdos141.erdos_141 Let $k≥3$ sorry line 78 source
Source documentation
Let $k≥3$. Are there $k$ consecutive primes in arithmetic progression?

depends on sorry what this means
Statement uses: Iff, True, Nat, GE.ge, OfNat.ofNat, Exists, Set, Erdos141.Set.IsAPAndPrimeProgressionOfLength
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_141 : answer(sorry) ↔
    ∀ k ≥ 3, ∃ (s : Set ℕ), s.IsAPAndPrimeProgressionOfLength k := by
…
```
theorem Erdos141.erdos_141.variants.first_cases The existence of such progressions has been verified for $k≤10$ sorry line 86 source
Source documentation
The existence of such progressions has been verified for $k≤10$.

depends on sorry what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, LE.le, Exists, Set, Erdos141.Set.IsAPAndPrimeProgressionOfLength
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_141.variants.first_cases :
    (∀ k ≥ 3, k ≤ 10 → ∃ (s : Set ℕ), s.IsAPAndPrimeProgressionOfLength k) := by
…
```
theorem Erdos141.erdos_141.variants.eleven Are there $11$ consecutive primes in arithmetic progression? sorry line 94 source
Source documentation
Are there $11$ consecutive primes in arithmetic progression?

depends on sorry what this means
Statement uses: Iff, True, Exists, Set, Nat, Erdos141.Set.IsAPAndPrimeProgressionOfLength, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_141.variants.eleven : answer(sorry) ↔
    ∃ (s : Set ℕ), s.IsAPAndPrimeProgressionOfLength 11 := by
…
```
def Erdos141.consecutivePrimeArithmeticProgressions The set of arithmetic progressions of consecutive primes of length $k$ line 101 source
Source documentation
The set of arithmetic progressions of consecutive primes of length $k$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set
Proof uses: Set.ofPred, Erdos141.Set.IsAPAndPrimeProgressionOfLength
```
def consecutivePrimeArithmeticProgressions (k : ℕ) : Set (Set ℕ) :=
…
```
theorem Erdos141.erdos_141.variants.infinite_three It is open, even for $k=3$, whether there are infinitely many such progressions sorry line 108 source
Source documentation
It is open, even for $k=3$, whether there are infinitely many such progressions.

depends on sorry what this means
Statement uses: Iff, True, Set.Infinite, Set, Nat, Erdos141.consecutivePrimeArithmeticProgressions, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_141.variants.infinite_three : answer(sorry) ↔
    (consecutivePrimeArithmeticProgressions 3).Infinite := by
…
```
Mentions: Erdos141.consecutivePrimeArithmeticProgressions
theorem Erdos141.erdos_141.variants.infinite_general_case Fix a $k \geq 3$ sorry line 116 source
Source documentation
Fix a $k \geq 3$. Is it true that there are infinitely many arithmetic prime progressions of length $k$?

depends on sorry what this means
Statement uses: Iff, True, Nat, GE.ge, OfNat.ofNat, Set.Infinite, Set, Erdos141.consecutivePrimeArithmeticProgressions
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_141.variants.infinite_general_case : answer(sorry) ↔
    ∀ k ≥ 3, (consecutivePrimeArithmeticProgressions k).Infinite := by
…
```
Mentions: Erdos141.consecutivePrimeArithmeticProgressions