18.lean`

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

12 declarations, 5 contain sorry, 0 declared as axioms

Erdős Problem 18

Reference:

erdosproblems.com/18
[ErGr80] Erdős, P. and Graham, R. L. (1980). Old and New Problems and Results in Combinatorial Number Theory. Monographies de L'Enseignement Mathématique, 28. Université de Genève. (See the sections on Egyptian fractions or practical numbers).
[Vo85] Vose, Michael D., Egyptian fractions. Bull. London Math. Soc. (1985), 21-24.

Imports (1)

FormalConjecturesUtil

def Erdos18.practicalH For a practical number $n$, $h(n)$ is the maximum over all $1 ≤ m ≤ n$ of the minimum number of divisors of $n$ needed to represent $m$ as a sum of distinct divisors line 40 source
Source documentation
For a practical number $n$, $h(n)$ is the maximum over all $1 ≤ m ≤ n$ of the minimum number of divisors of $n$ needed to represent $m$ as a sum of distinct divisors.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.sup, Finset.Icc, OfNat.ofNat, InfSet.sInf, Set.ofPred, Exists, Finset, And, LE.le, Nat.divisors, Eq, Finset.card, Membership.mem, Set, subsetSums
14 instances, coercions and auxiliary constants left out.
```
noncomputable def practicalH (n : ℕ) : ℕ :=
…
```
theorem Erdos18.practicalH_one $h(1) = 1$: we need the single divisor {1} to represent 1 line 48 source
Source documentation
$h(1) = 1$: we need the single divisor {1} to represent 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Erdos18.practicalH, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, Finset.sup, Finset.Icc, InfSet.sInf, Set.ofPred, Exists, Finset, And, LE.le, Nat.divisors, Finset.card, Membership.mem, Set, subsetSums, Or, Singleton.singleton and 36 more
39 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem practicalH_one : practicalH 1 = 1 := by
…
```
Mentions: Erdos18.practicalH
theorem Erdos18.practicalH_two $h(2) = 1$: divisors are {1, 2}, each of m=1,2 needs only 1 divisor line 53 source
Source documentation
$h(2) = 1$: divisors are {1, 2}, each of m=1,2 needs only 1 divisor.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Erdos18.practicalH, OfNat.ofNat
Proof uses: Eq.mpr, Max.max, InfSet.sInf, Set.ofPred, Exists, Finset, And, LE.le, Insert.insert, Singleton.singleton, Finset.card, Membership.mem, Set, subsetSums, id, congrFun', congrArg, Eq.trans, Finset.sup, Finset.Icc and 37 more
56 instances, coercions and auxiliary constants left out.

Tactic trace: 16 steps
Open to load the checked before-and-after goal states.
```
theorem practicalH_two : practicalH 2 = 1 := by
…
```
Mentions: Erdos18.practicalH
theorem Erdos18.practicalH_six $h(6) = 2$: divisors are {1, 2, 3, 6} line 69 source
Source documentation
$h(6) = 2$: divisors are {1, 2, 3, 6}. The hardest m to represent is m=4 or m=5, each requiring 2 divisors: 4=1+3, 5=2+3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Erdos18.practicalH, OfNat.ofNat
Proof uses: Finset, Nat.divisors, Insert.insert, Singleton.singleton, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, le_antisymm, Finset.sup_le, Finset.Icc, InfSet.sInf, Set.ofPred, Exists, And, LE.le, Finset.card, Membership.mem and 61 more
66 instances, coercions and auxiliary constants left out.

Tactic trace: 84 steps
Open to load the checked before-and-after goal states.
```
theorem practicalH_six : practicalH 6 = 2 := by
…
```
Mentions: Erdos18.practicalH
theorem Erdos18.practicalH_twelve $h(12) = 3$: divisors are {1, 2, 3, 4, 6, 12} line 110 source
Source documentation
$h(12) = 3$: divisors are {1, 2, 3, 4, 6, 12}. The hardest m is m=11, requiring 3 divisors: 11=1+4+6.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Erdos18.practicalH, OfNat.ofNat
Proof uses: Finset, Nat.divisors, Insert.insert, Singleton.singleton, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, le_antisymm, Finset.sup_le, Finset.Icc, InfSet.sInf, Set.ofPred, Exists, And, LE.le, Finset.card, Membership.mem and 71 more
70 instances, coercions and auxiliary constants left out.

Tactic trace: 140 steps
Open to load the checked before-and-after goal states.
```
theorem practicalH_twelve : practicalH 12 = 3 := by
…
```
Mentions: Erdos18.practicalH
theorem Erdos18.practicalH_le_divisors For any practical number $n$, $h(n)$ ≤ number of divisors of $n$ line 162 source
Source documentation
For any practical number $n$, $h(n)$ ≤ number of divisors of $n$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Nat.IsPractical, LE.le, Erdos18.practicalH, Finset.card, Nat.divisors
Proof uses: Eq.mpr, And, OfNat.ofNat, InfSet.sInf, Set.ofPred, Exists, Finset, Eq, Membership.mem, Set, subsetSums, id, Eq.trans, Finset.sup, Finset.Icc, forall_congr, implies_congr, Eq.refl, Nat.sInf_le, Exists.intro and 3 more
21 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem practicalH_le_divisors (n : ℕ) (hn : Nat.IsPractical n) :
    practicalH n ≤ n.divisors.card := by
…
```
Mentions: Nat.IsPractical, Erdos18.practicalH
theorem Erdos18.factorial_isPractical $h(n!)$ is well-defined since $n!$ is practical for $n ≥ 1$ line 169 source
Source documentation
$h(n!)$ is well-defined since $n!$ is practical for $n ≥ 1$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Nat.IsPractical, Nat.factorial
Proof uses: Nat.recAux, LE.le, OfNat.ofNat, dite, Membership.mem, Set, subsetSums, Finset, Nat.divisors, Exists.intro, And, Eq, Finset.sum, Singleton.singleton, And.intro, of_eq_true, Eq.trans, True, congr, congrArg and 123 more
141 instances, coercions and auxiliary constants left out.

Tactic trace: 64 steps
Open to load the checked before-and-after goal states.
```
theorem factorial_isPractical (n : ℕ) : Nat.IsPractical n.factorial := by
…
```
Mentions: Nat.IsPractical
theorem Erdos18.erdos_18a Conjecture 1 sorry line 221 source
Source documentation
Conjecture 1. Are there infinitely many practical numbers $m$ such that $h(m) < (\log \log m)^{O(1)}$?
More precisely: does there exist a constant $C > 0$ such that for infinitely many practical numbers $m$, we have $h(m) < (\log \log m)^C$?

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, LT.lt, OfNat.ofNat, Filter.Frequently, Nat, Nat.IsPractical, Erdos18.practicalH, HPow.hPow, Real.log, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_18a : answer(sorry) ↔
    ∃ C : ℝ, 0 < C ∧ ∃ᶠ m in atTop, Nat.IsPractical m ∧
      (practicalH m : ℝ) < (log (log m)) ^ C := by
…
```
Mentions: Nat.IsPractical, Erdos18.practicalH
theorem Erdos18.erdos_18b Conjecture 2 sorry line 232 source
Source documentation
Conjecture 2. Is it true that $h(n!) < n^{o(1)}$? That is, for all $\varepsilon > 0$, is $h(n!) < n^\varepsilon$ for sufficiently large $n$?

depends on sorry what this means
Statement uses: Iff, True, Real, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, Erdos18.practicalH, Nat.factorial, HPow.hPow, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_18b : answer(sorry) ↔
    ∀ ε : ℝ, 0 < ε → ∀ᶠ n : ℕ in atTop, (practicalH n.factorial : ℝ) < (n : ℝ) ^ ε := by
…
```
Mentions: Erdos18.practicalH
theorem Erdos18.erdos_18c Conjecture 3 sorry line 243 source
Source documentation
Conjecture 3. Or perhaps even $h(n!) < (\log n)^{O(1)}$?
Erdős offered $250 for a proof or disproof.

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, Erdos18.practicalH, Nat.factorial, HPow.hPow, Real.log, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_18c : answer(sorry) ↔
    ∃ C : ℝ, 0 < C ∧ ∀ᶠ n : ℕ in atTop, (practicalH n.factorial : ℝ) < (log n) ^ C := by
…
```
Mentions: Erdos18.practicalH
theorem Erdos18.erdos_18_upper_bound Erdős's Theorem sorry line 252 source
Source documentation
Erdős's Theorem. Erdős proved that $h(n!) < n$ for all $n \ge 1$.

depends on sorry what this means
Statement uses: Filter.Eventually, Nat, LT.lt, Erdos18.practicalH, Nat.factorial, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_18_upper_bound :
    ∀ᶠ n : ℕ in atTop, practicalH (Nat.factorial n) < n := by
…
```
Mentions: Erdos18.practicalH
theorem Erdos18.erdos_18_vose Vose's Theorem sorry line 262 source
Source documentation
Vose's Theorem. Vose proved the existence of infinitely many practical numbers $m$ such that $h(m) \ll (\log m)^{1/2}$. This gives a positive answer to a weaker form of Conjecture 1.

depends on sorry what this means
Statement uses: Exists, Real, And, LT.lt, OfNat.ofNat, Filter.Frequently, Nat, Nat.IsPractical, Erdos18.practicalH, HMul.hMul, HPow.hPow, Real.log, HDiv.hDiv, Filter.atTop
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_18_vose :
    ∃ C : ℝ, 0 < C ∧ ∃ᶠ m in atTop, Nat.IsPractical m ∧
      (practicalH m : ℝ) < C * (log m) ^ (1 / 2 : ℝ) := by
…
```
Mentions: Nat.IsPractical, Erdos18.practicalH