342.lean`

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

9 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 342

References:

erdosproblems.com/342
Ben Green's Open Problem 7
OEIS A002858
[Gu04] Guy, Richard K., Unsolved problems in number theory (2004), xviii+437.

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/GreensOpenProblems/7.lean

def Erdos342.UniqueUlamSum UniqueUlamSum a n m means that $m$ has a unique representation as $a(i) + a(j)$ with $i < j < n$ line 39 source
Source documentation
UniqueUlamSum a n m means that $m$ has a unique representation as $a(i) + a(j)$ with $i < j < n$.

no axioms what this means
Statement uses: Nat
Proof uses: ExistsUnique, Prod, And, LT.lt, Prod.fst, Prod.snd, Eq, HAdd.hAdd
3 instances, coercions and auxiliary constants left out.
```
def UniqueUlamSum (a : ℕ → ℕ) (n m : ℕ) : Prop :=
…
```
def Erdos342.IsUlamSequence Documented declaration: IsUlamSequence line 46 source
Source documentation
IsUlamSequence a means that $a$ is the Ulam sequence (OEIS A002858): $a(0) = 1$, $a(1) = 2$, and for each $n \geq 2$, $a(n)$ is the least integer greater than $a(n-1)$ that has a unique representation as $a(i) + a(j)$ with $i < j < n$.

no axioms what this means
Statement uses: Nat
Proof uses: And, Eq, OfNat.ofNat, LE.le, LT.lt, HSub.hSub, Erdos342.UniqueUlamSum, Not
5 instances, coercions and auxiliary constants left out.
```
def IsUlamSequence (a : ℕ → ℕ) : Prop :=
…
```
theorem Erdos342.erdos_342.test.a0 $a(0) = 1$ by definition line 55 source
Source documentation
$a(0) = 1$ by definition.

no axioms what this means
Statement uses: Nat, Erdos342.IsUlamSequence, Eq, OfNat.ofNat
Proof uses: LE.le, And, LT.lt, HSub.hSub, Erdos342.UniqueUlamSum, Not
7 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_342.test.a0 : ∀ a : ℕ → ℕ, IsUlamSequence a → a 0 = 1 := by
…
```
Mentions: Erdos342.IsUlamSequence
theorem Erdos342.erdos_342.test.a1 $a(1) = 2$ by definition line 60 source
Source documentation
$a(1) = 2$ by definition.

no axioms what this means
Statement uses: Nat, Erdos342.IsUlamSequence, Eq, OfNat.ofNat
Proof uses: LE.le, And, LT.lt, HSub.hSub, Erdos342.UniqueUlamSum, Not
7 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_342.test.a1 : ∀ a : ℕ → ℕ, IsUlamSequence a → a 1 = 2 := by
…
```
Mentions: Erdos342.IsUlamSequence
theorem Erdos342.erdos_342.test.a2 $a(2) = 3$: the only pair $(i,j)$ with $i < j < 2$ is $(0,1)$, giving $1 + 2 = 3$ line 65 source
Source documentation
$a(2) = 3$: the only pair $(i,j)$ with $i < j < 2$ is $(0,1)$, giving $1 + 2 = 3$.

standard axioms: Quot.sound, propext what this means
Statement uses: Nat, Erdos342.IsUlamSequence, Eq, OfNat.ofNat
Proof uses: LE.le, And, LT.lt, HSub.hSub, Erdos342.UniqueUlamSum, Not, Decidable.byContradiction, Prod, Prod.fst, Prod.snd, HAdd.hAdd, Prod.mk, dite, of_eq_true, Eq.trans, True, congrFun', congrArg, id, Eq.mp and 13 more
20 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_342.test.a2 : ∀ a : ℕ → ℕ, IsUlamSequence a → a 2 = 3 := by
…
```
Mentions: Erdos342.IsUlamSequence
theorem Erdos342.erdos_342.test.a3 $a(3) = 4$: among sums $> 3$ with a unique representation from $\{1,2,3\}$, the smallest is $4 = 1 + 3$ line 74 source
Source documentation
$a(3) = 4$: among sums $> 3$ with a unique representation from $\{1,2,3\}$, the smallest is $4 = 1 + 3$. The candidate $5 = 2 + 3$ is ruled out by minimality since $4$ has a unique representation.

standard axioms: Quot.sound, propext what this means
Statement uses: Nat, Erdos342.IsUlamSequence, Eq, OfNat.ofNat
Proof uses: LE.le, And, LT.lt, HSub.hSub, Erdos342.UniqueUlamSum, Not, Erdos342.erdos_342.test.a2, And.intro, Decidable.byContradiction, Prod, Prod.fst, Prod.snd, HAdd.hAdd, Prod.mk, dite, False.elim, Eq.mp, forall_congr, implies_congr, congrFun' and 15 more
31 instances, coercions and auxiliary constants left out.

Tactic trace: 51 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_342.test.a3 : ∀ a : ℕ → ℕ, IsUlamSequence a → a 3 = 4 := by
…
```
Mentions: Erdos342.IsUlamSequence
theorem Erdos342.erdos_342.parts.i Do infinitely many pairs $(a, a+2)$ occur in Ulam's sequence? sorry line 115 source
Source documentation
Do infinitely many pairs $(a, a+2)$ occur in Ulam's sequence?

depends on sorry what this means
Statement uses: Iff, True, Nat, Erdos342.IsUlamSequence, Set.Infinite, Set.ofPred, Exists, Eq, HAdd.hAdd, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_342.parts.i :
    answer(sorry) ↔
      ∀ a : ℕ → ℕ, IsUlamSequence a →
        Set.Infinite {n : ℕ | ∃ m, a m = a n + 2} := by
…
```
Mentions: Erdos342.IsUlamSequence
theorem Erdos342.erdos_342.parts.ii Does Ulam's sequence eventually have periodic differences? sorry line 125 source
Source documentation
Does Ulam's sequence eventually have periodic differences? That is, is $a(n+1) - a(n)$ eventually periodic?

depends on sorry what this means
Statement uses: Iff, True, Nat, Erdos342.IsUlamSequence, Int, HSub.hSub, HAdd.hAdd, OfNat.ofNat, Exists, And, GT.gt, Filter.Eventually, Eq, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_342.parts.ii :
    answer(sorry) ↔
      ∀ a : ℕ → ℕ, IsUlamSequence a →
        let d (n : ℕ) : ℤ := a (n + 1) - a n
…
```
Mentions: Erdos342.IsUlamSequence
theorem Erdos342.erdos_342.parts.iii Part (iii), is the density of the sequence 0? sorry line 136 source
Source documentation
Part (iii), is the density of the sequence 0?

depends on sorry what this means
Statement uses: Iff, True, Nat, Erdos342.IsUlamSequence, Eq, Real, Set.upperDensity, Set.range, Set.univ, OfNat.ofNat
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 erdos_342.parts.iii :
    answer(sorry) ↔
      ∀ a : ℕ → ℕ, IsUlamSequence a →
        Set.upperDensity (Set.range a) = 0 := by
…
```
Mentions: Erdos342.IsUlamSequence, Set.upperDensity