308.lean`

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

9 declarations, 5 contain sorry, 0 declared as axioms

Erdős Problem 308

References:

erdosproblems.com/308
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
[Cr99] Croot, III, Ernest S., On some questions of Erdős and Graham about Egyptian fractions. Mathematika (1999), 359-372.

Imports (1)

FormalConjecturesUtil

def Erdos308.IsRepresentable k is representable as a sum of distinct unit fractions with denominators from {1, …, N} line 39 source
Source documentation
k is representable as a sum of distinct unit fractions with denominators from {1, …, N}.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Exists, Finset, And, LE.le, Finset.Icc, OfNat.ofNat, Eq, Rat, Finset.sum, HDiv.hDiv
12 instances, coercions and auxiliary constants left out.
```
def IsRepresentable (N k : ℕ) : Prop :=
…
```
def Erdos308.representable The set of positive integers representable as a sum of distinct unit fractions with denominators from {1, …, N} line 44 source
Source documentation
The set of positive integers representable as a sum of distinct unit fractions with denominators from {1, …, N}.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set
Proof uses: Set.ofPred, And, LT.lt, OfNat.ofNat, Erdos308.IsRepresentable
2 instances, coercions and auxiliary constants left out.
```
def representable (N : ℕ) : Set ℕ := {k | 0 < k ∧ IsRepresentable N k}
```
Mentions: Erdos308.IsRepresentable
def Erdos308.f f N is the smallest positive integer which is not representable as a sum of distinct unit fractions with denominators from {1, …, N} line 48 source
Source documentation
f N is the smallest positive integer which is not representable as a sum of distinct unit fractions with denominators from {1, …, N}.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: InfSet.sInf, Set.ofPred, And, LT.lt, OfNat.ofNat, Not, Erdos308.IsRepresentable
3 instances, coercions and auxiliary constants left out.
```
noncomputable def f (N : ℕ) : ℕ := sInf {k | 0 < k ∧ ¬ IsRepresentable N k}
```
Mentions: Erdos308.IsRepresentable
def Erdos308.m m N = ⌊∑_{n ≤ N} 1/n⌋ line 51 source
Source documentation
m N = ⌊∑_{n ≤ N} 1/n⌋.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Nat.floor, Rat, harmonic
7 instances, coercions and auxiliary constants left out.
```
def m (N : ℕ) : ℕ := ⌊harmonic N⌋₊
…
```
theorem Erdos308.erdos_308.parts.i sorry line 63 source
depends on sorry what this means
Statement uses: Filter.Eventually, Nat, Or, Eq, Erdos308.f, Erdos308.m, HAdd.hAdd, OfNat.ofNat, Filter.atTop
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 Erdos308.erdos_308.parts.ii sorry line 74 source
depends on sorry what this means
Statement uses: Iff, True, Filter.Eventually, Nat, Exists, Eq, Set, Erdos308.representable, Set.Icc, OfNat.ofNat, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos308.erdos_308.variants.shape sorry line 84 source
depends on sorry what this means
Statement uses: Filter.Eventually, Nat, Or, Eq, Set, Erdos308.representable, Set.Icc, OfNat.ofNat, HSub.hSub, Erdos308.m, Filter.atTop
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 Erdos308.erdos_308.variants.croot Croot [Cr99] bounded sorry line 98 source
Source documentation
Croot [Cr99] bounded $\eta(N)$, the largest integer such that every integer in $\{1,\ldots,\eta(N)\}$ is representable; this is $f(N)-1$, so his bounds read $$\left\lfloor\sum_{n\leq N}\frac{1}{n}-\frac{9}{2}(1+o(1))\frac{(\log\log N)^2}{\log N} \right\rfloor + 1 \leq f(N)$$ and $$f(N)\leq \left\lfloor\sum_{n\leq N}\frac{1}{n}-\frac{1}{2}(1+o(1)) \frac{(\log\log N)^2}{\log N}\right\rfloor + 1.$$

depends on sorry what this means
Statement uses: Exists, Nat, Real, And, Filter.Tendsto, Filter.atTop, nhds, OfNat.ofNat, Filter.Eventually, LE.le, HAdd.hAdd, Nat.floor, HSub.hSub, harmonic, HDiv.hDiv, HMul.hMul, HPow.hPow, Real.log, Erdos308.f
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
74 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_308.variants.croot :
    ∃ o₁ o₂ : ℕ → ℝ, Tendsto o₁ atTop (nhds 0) ∧ Tendsto o₂ atTop (nhds 0) ∧
      ∀ᶠ N : ℕ in atTop,
        ⌊(harmonic N : ℝ) - 9 / 2 * (1 + o₁ N) * (log (log N)) ^ 2 / log N⌋₊ + 1 ≤ f N ∧
        f N ≤ ⌊(harmonic N : ℝ) - 1 / 2 * (1 + o₂ N) * (log (log N)) ^ 2 / log N⌋₊ + 1 := by
…
```
Mentions: Erdos308.f
theorem Erdos308.erdos_308.variants.all_N Documented declaration: all_N sorry line 111 source
Source documentation
Is it true that, for every $N\geq 1$, the set of integers representable as the sum of distinct unit fractions with denominators from $\{1,\ldots,N\}$ has the shape $\{1,\ldots,m\}$ for some $m$?

depends on sorry what this means
Statement uses: Iff, True, Nat, GE.ge, OfNat.ofNat, Exists, Eq, Set, Erdos308.representable, Set.Icc
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_308.variants.all_N : answer(sorry) ↔
    ∀ N ≥ 1, ∃ m, representable N = Set.Icc 1 m := by
…
```
Mentions: Erdos308.m, Erdos308.representable