647.lean`

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4 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 647

Reference: erdosproblems.com/647

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100OpenSet1.lean

theorem Erdos647.erdos_647 Let $\tau(n)$ count the number of divisors of $n$ sorry line 37 source
Source documentation
Let $\tau(n)$ count the number of divisors of $n$. Is there some $n > 24$ such that $$ \max_{m < n}(m + \tau(m)) \leq n + 2? $$

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, And, GT.gt, OfNat.ofNat, LE.le, iSup, Fin, HAdd.hAdd, ArithmeticFunction, ArithmeticFunction.sigma
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_647 : answer(sorry) ↔ ∃ n > 24, ⨆ m : Fin n, m + σ 0 m ≤ n + 2 := by
…
```
theorem Erdos647.erdos_647.variants.twenty_four This is true for $n = 24$ line 42 source
Source documentation
This is true for $n = 24$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: LE.le, Nat, iSup, Fin, OfNat.ofNat, HAdd.hAdd, ArithmeticFunction, ArithmeticFunction.sigma
Proof uses: ciSup_le, of_decide_eq_true, id, Eq, Bool, Decidable.decide, Bool.true, Eq.refl
31 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_647.variants.twenty_four : ⨆ m : Fin 24, (m : ℕ) + σ 0 m ≤ 26 := by
…
```
theorem Erdos647.erdos_647.variants.lim Erdős says 'it is extremely doubtful' that there are infinitely many such $n$, and in fact suggests that $$ lim_{n\to\infty} \max_{m < n}(\tau(m) + m − n) = \infty. $$ sorry line 51 source
Source documentation
Erdős says 'it is extremely doubtful' that there are infinitely many such $n$, and in fact suggests that $$ lim_{n\to\infty} \max_{m < n}(\tau(m) + m − n) = \infty. $$

depends on sorry what this means
Statement uses: Iff, True, Filter.Tendsto, Nat, iSup, Fin, HSub.hSub, HAdd.hAdd, ArithmeticFunction, ArithmeticFunction.sigma, OfNat.ofNat, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_647.variants.lim :
    answer(sorry) ↔ atTop.Tendsto (fun n ↦ ⨆ m : Fin n, σ 0 m + m - n) atTop := by
…
```
theorem Erdos647.erdos_647.variants.infinite Erdős says it 'seems certain' that for every $k$ there are infinitely many $n$ for which $$ \max_{n−k < m < n}(m + \tau(m)) ≤ n + 2. $$ sorry line 61 source
Source documentation
Erdős says it 'seems certain' that for every $k$ there are infinitely many $n$ for which $$ \max_{n−k < m < n}(m + \tau(m)) ≤ n + 2. $$

depends on sorry what this means
Statement uses: Iff, True, Nat, Set.Infinite, Set.ofPred, LE.le, iSup, Set.Ioo, HSub.hSub, HAdd.hAdd, Membership.mem, Set, ArithmeticFunction, ArithmeticFunction.sigma, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_647.variants.infinite :
    answer(sorry) ↔ ∀ k, { n | ⨆ m : Set.Ioo (n - k) n, ↑m + σ 0 m ≤ n + 2 }.Infinite := by
…
```