1101.lean`

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

6 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 1101

Reference: erdosproblems.com/1101

Imports (1)

FormalConjecturesUtil

def Erdos1101.ASet The set of integers not divisible by any u_i line 33 source
Source documentation
The set of integers not divisible by any u_i.

no axioms what this means
Statement uses: Nat, Set
Proof uses: Set.ofPred, Not, Dvd.dvd
1 instances, coercions and auxiliary constants left out.
```
def ASet (u : ℕ → ℕ) : Set ℕ :=
…
```
def Erdos1101.A The sequence of integers A_u which are not divisible by any u_i arranged in a monotonic sequence line 38 source
Source documentation
The sequence of integers A_u which are not divisible by any u_i arranged in a monotonic sequence.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Nat.nth, Membership.mem, Set, Erdos1101.ASet
1 instances, coercions and auxiliary constants left out.
```
noncomputable def A (u : ℕ → ℕ) (n : ℕ) : ℕ :=
…
```
def Erdos1101.t t_x such that u_0 line 43 source
Source documentation
t_x such that u_0 ... u_{t_x-1} ≤ x < u_0 ... u_{t_x}.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SupSet.sSup, Set.ofPred, LE.le, Finset.prod, Finset.range
3 instances, coercions and auxiliary constants left out.
```
noncomputable def t (u : ℕ → ℕ) (x : ℕ) : ℕ :=
…
```
def Erdos1101.IsGood A sequence is "good" if 1 line 51 source
Source documentation
A sequence is "good" if
1. it is strictly monotone
2. it is pairwise coprime
3. the sum of reciprocals converges
4. the gap between consecutive elements in A(u) is bounded relative to t_x.
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: And, StrictMono, Ne, Nat.Coprime, Summable, Real, HDiv.hDiv, OfNat.ofNat, SummationFilter.unconditional, GT.gt, Filter.Eventually, LT.lt, Erdos1101.A, HSub.hSub, HAdd.hAdd, HMul.hMul, Erdos1101.t, Inv.inv, tprod, Filter.atTop
26 instances, coercions and auxiliary constants left out.
```
def IsGood (u : ℕ → ℕ) : Prop :=
…
```
theorem Erdos1101.erdos_1101.parts.i 1 sorry line 61 source
Source documentation
1. There is NO good sequence with polynomial growth.
depends on sorry what this means
Statement uses: Not, Exists, Nat, And, Erdos1101.IsGood, Asymptotics.IsBigO, Real, Filter.atTop, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1101.parts.i :
    ¬ ∃ u, IsGood u ∧ ∃ k : ℕ, (fun n => (u n : ℝ)) =O[atTop] (fun n => (n : ℝ) ^ k) := by
…
```
Mentions: Erdos1101.IsGood
theorem Erdos1101.erdos_1101.parts.ii 2 sorry line 67 source
Source documentation
1. There is a good sequence with sub-exponential growth.
depends on sorry what this means
Statement uses: Exists, Nat, And, Erdos1101.IsGood, Asymptotics.IsLittleO, Real, Filter.atTop, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1101.parts.ii :
    ∃ u, IsGood u ∧ (fun n => Real.log (u n : ℝ)) =o[atTop] (fun n => (n : ℝ)) := by
…
```
Mentions: Erdos1101.IsGood