261.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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

9 declarations, 7 contain sorry, 0 declared as axioms

Erdős Problem 261

References:

erdosproblems.com/261
[BoLo90] Borwein, Peter and Loring, Terry A., Some questions of Erdős and Graham on numbers of the form $\sum g_n/2^{g_n}$. Math. Comp. (1990), 377--394.
[Er88c] Erdős, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102--109.
[TUZ20] Tengely, Szabolcs and Ulas, Maciej and Zygadlo, Jakub, On a Diophantine equation of Erdős and Graham. J. Number Theory (2020), 445--459.

Imports (1)

FormalConjecturesUtil

def Erdos261.Erdos261Prop A natural number line 42 source
Source documentation
A natural number $n$ is said to have property Erdos261Prop if there exist $t \ge 2$ pairwise distinct positive integers $a_1, \ldots, a_t$ such that $n / 2^n = \sum_{1 \le k \le t} a_k / 2^{a_k}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Exists, And, GE.ge, OfNat.ofNat, Fin, Function.Injective, LE.le, Eq, Rat, HDiv.hDiv, HPow.hPow, Finset.sum, Finset.univ
15 instances, coercions and auxiliary constants left out.
```
def Erdos261Prop (n : ℕ) : Prop := ∃ᵉ (t ≥ 2) (a : Fin t → ℕ), a.Injective ∧
…
```
def Erdos261.Erdos261InfiniteRepresentation A canonical infinite representation of a rational number $x$ by positive integers line 48 source
Source documentation
A canonical infinite representation of a rational number $x$ by positive integers. The denominators are strictly increasing so that reorderings are not counted as different representations.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Rat, Nat
Proof uses: And, StrictMono, LE.le, OfNat.ofNat, Summable, HDiv.hDiv, HPow.hPow, SummationFilter.unconditional, Eq, tsum
21 instances, coercions and auxiliary constants left out.
```
def Erdos261InfiniteRepresentation (x : ℚ) (a : ℕ → ℕ) : Prop :=
…
```
theorem Erdos261.erdos_261.variants.borwein_loring For every positive integer $m$, if $n = 2^{m+1} - m - 2$, then $$\frac{n}{2^n} = \sum_{n < k \le n + m} \frac{k}{2^k}.$$ This construction is due to Borwein and Loring [BoLo90] sorry line 57 source
Source documentation
For every positive integer $m$, if $n = 2^{m+1} - m - 2$, then $$\frac{n}{2^n} = \sum_{n < k \le n + m} \frac{k}{2^k}.$$
This construction is due to Borwein and Loring [BoLo90].

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, HSub.hSub, HPow.hPow, HAdd.hAdd, Eq, Rat, HDiv.hDiv, Finset.sum, Finset.Ioc
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_261.variants.borwein_loring (m : ℕ) (hm : 0 < m) :
    let n := 2 ^ (m + 1) - m - 2
…
```
theorem Erdos261.erdos_261.variants.borwein_loring_property The Borwein--Loring construction gives the required property when $m \ge 2$ sorry line 65 source
Source documentation
The Borwein--Loring construction gives the required property when $m \ge 2$. This lower bound ensures that the representation contains at least two terms.

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Erdos261.Erdos261Prop, HSub.hSub, HPow.hPow, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_261.variants.borwein_loring_property (m : ℕ) (hm : 2 ≤ m) :
    Erdos261Prop (2 ^ (m + 1) - m - 2) := by
…
```
Mentions: Erdos261.Erdos261Prop
theorem Erdos261.erdos_261.parts.i Are there infinitely many positive integers sorry line 75 source
Source documentation
Are there infinitely many positive integers $n$ such that there exist some $t \ge 2$ and distinct integers $a_1, \ldots, a_t \ge 1$ satisfying $$\frac{n}{2^n} = \sum_{1 \le k \le t} \frac{a_k}{2^{a_k}}?$$
In [Er88c], Erdős notes that Cusick had a simple proof that infinitely many such $n$ exist.

depends on sorry what this means
Statement uses: Iff, True, Set.Infinite, Nat, Set.ofPred, And, LT.lt, OfNat.ofNat, Erdos261.Erdos261Prop
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 erdos_261.parts.i : answer(True) ↔ {n : ℕ | 0 < n ∧ Erdos261Prop n}.Infinite := by
…
```
Mentions: Erdos261.Erdos261Prop
theorem Erdos261.erdos_261.variants.le_10000 Tengely, Ulas, and Zygadlo [TUZ20] verified that every positive integer $n \le 10000$ has the required property sorry line 81 source
Source documentation
Tengely, Ulas, and Zygadlo [TUZ20] verified that every positive integer $n \le 10000$ has the required property.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, LE.le, Erdos261.Erdos261Prop
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_261.variants.le_10000 {n : ℕ} (hn_pos : 0 < n) (hn : n ≤ 10000) :
    Erdos261Prop n := by
…
```
Mentions: Erdos261.Erdos261Prop
theorem Erdos261.erdos_261.parts.ii Do all positive integers $n$ have the required property? sorry line 87 source
Source documentation
Do all positive integers $n$ have the required property?

depends on sorry what this means
Statement uses: Iff, True, Nat, GT.gt, OfNat.ofNat, Erdos261.Erdos261Prop
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 erdos_261.parts.ii : answer(sorry) ↔ ∀ n > 0, Erdos261Prop n := by
…
```
Mentions: Erdos261.Erdos261Prop
theorem Erdos261.erdos_261.parts.iii Is there a rational number $x$ such that $$x = \sum_{k=1}^{\infty} \frac{a_k}{2^{a_k}}$$ has at least $2^{\aleph_0}$ representations by pairwise distinct positive integers $a_k$? sorry line 94 source
Source documentation
Is there a rational number $x$ such that $$x = \sum_{k=1}^{\infty} \frac{a_k}{2^{a_k}}$$ has at least $2^{\aleph_0}$ representations by pairwise distinct positive integers $a_k$?

depends on sorry what this means
Statement uses: Iff, True, Exists, Rat, LE.le, Cardinal, Cardinal.continuum, Cardinal.mk, Nat, Set.ofPred, Erdos261.Erdos261InfiniteRepresentation
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_261.parts.iii : answer(sorry) ↔ ∃ x : ℚ,
    𝔠 ≤ #{a : ℕ → ℕ | Erdos261InfiniteRepresentation x a} := by
…
```
Mentions: Erdos261.Erdos261InfiniteRepresentation
theorem Erdos261.erdos_261.variants.two_representations sorry line 109 source
depends on sorry what this means
Statement uses: Iff, True, Exists, Rat, LE.le, Cardinal, OfNat.ofNat, Nat, Cardinal.mk, Set.ofPred, Erdos261.Erdos261InfiniteRepresentation
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.