359.lean`

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

5 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 359

Reference: erdosproblems.com/359

Imports (1)

FormalConjecturesUtil

def Erdos359.IsGoodFor The predicate that A is monotone, A 0 = n and for all j, A (j + 1) is the smallest natural number that cannot be written as a sum of consecutive terms of A 0, ..., A j line 34 source
Source documentation
The predicate that A is monotone, A 0 = n and for all j, A (j + 1) is the smallest natural number that cannot be written as a sum of consecutive terms of A 0, ..., A j

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: And, Eq, OfNat.ofNat, StrictMono, IsLeast, Set.ofPred, LT.lt, LE.le, Finset, Finset.Icc, Finset.Iic, Ne, Finset.sum, HAdd.hAdd
13 instances, coercions and auxiliary constants left out.
```
def IsGoodFor (A : ℕ → ℕ) (n : ℕ) : Prop := A 0 = n ∧ StrictMono A ∧
…
```
theorem Erdos359.erdos_359.parts.i Let $a_1< a_2 < ⋯ $ be an infinite sequence of integers such that $a_1=1$ and $a_{i+1}$ is the least integer which is not a sum of consecutive earlier $a_j$s sorry line 42 source
Source documentation
Let $a_1< a_2 < ⋯ $ be an infinite sequence of integers such that $a_1=1$ and $a_{i+1}$ is the least integer which is not a sum of consecutive earlier $a_j$s. Show that $a_k / k \to \infty$.

depends on sorry what this means
Statement uses: Nat, Erdos359.IsGoodFor, OfNat.ofNat, Filter.Tendsto, Real, HDiv.hDiv, Filter.atTop
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_359.parts.i (A : ℕ → ℕ) (hA : IsGoodFor A 1) :
    atTop.Tendsto (fun k ↦ (A k : ℝ) / k) atTop := by
…
```
Mentions: Erdos359.IsGoodFor
theorem Erdos359.erdos_359.parts.ii Let $a_1< a_2 < ⋯ $ be an infinite sequence of integers such that $a_1=1$ and $a_{i+1}$ is the least integer which is not a sum of consecutive earlier $a_j$s sorry line 50 source
Source documentation
Let $a_1< a_2 < ⋯ $ be an infinite sequence of integers such that $a_1=1$ and $a_{i+1}$ is the least integer which is not a sum of consecutive earlier $a_j$s. Show that $a_k / k ^ {1 + c} \to 0$ for any $c > 0$.

depends on sorry what this means
Statement uses: Nat, Erdos359.IsGoodFor, OfNat.ofNat, Real, LT.lt, Filter.Tendsto, HDiv.hDiv, HPow.hPow, HAdd.hAdd, Filter.atTop, nhds
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_359.parts.ii (A : ℕ → ℕ) (hA : IsGoodFor A 1) (c : ℝ) (hc : 0 < c):
    atTop.Tendsto (fun k ↦ A k / (k : ℝ) ^ (1 + c)) (nhds 0) := by
…
```
Mentions: Erdos359.IsGoodFor, c
theorem Erdos359.erdos_359.variants.isGoodFor_1_low_values Suppose monotone sequence sorry line 58 source
Source documentation
Suppose monotone sequence $A$ satisfies the following: A 0 = 1 and for all j, A (j + 1) is the smallest natural number that cannot be written as a sum of consecutive terms of A 0, ..., A j. Then the first few terms of $A$ are $1,2,4,5,8,10,14,15,...$.

depends on sorry what this means
Statement uses: Nat, Erdos359.IsGoodFor, OfNat.ofNat, Eq, Set, Set.image, Set.Iic, Insert.insert, Singleton.singleton
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
8 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_359.variants.isGoodFor_1_low_values (A : ℕ → ℕ) (hA : IsGoodFor A 1) :
    A '' (Set.Iic 7) = {1, 2, 4, 5, 8, 10, 14, 15} := by
…
```
Mentions: Erdos359.IsGoodFor
theorem Erdos359.erdos_359.variants.isGoodFor_1_asymptotic Suppose monotone sequence sorry line 66 source
Source documentation
Suppose monotone sequence $A$ satisfies the following: A 0 = 1 and for all j, A (j + 1) is the smallest natural number that cannot be written as a sum of consecutive terms of A 0, ..., A j. Then it is conjectured that $$a_k ~ \frac{k \log k}{\log \log k}$$.

depends on sorry what this means
Statement uses: Nat, Erdos359.IsGoodFor, OfNat.ofNat, Asymptotics.IsEquivalent, Real, Filter.atTop, HDiv.hDiv, HMul.hMul, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_359.variants.isGoodFor_1_asymptotic (A : ℕ → ℕ) (hA : IsGoodFor A 1) :
    (fun k ↦ (A k : ℝ)) ~[atTop] (fun k ↦ k * (k : ℝ).log / (k : ℝ).log.log) := by
…
```
Mentions: Erdos359.IsGoodFor