247.lean`

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

2 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 247

Reference: erdosproblems.com/247

Imports (1)

FormalConjecturesUtil

theorem Erdos247.erdos_247 Let $n_1 < n_2 < \cdots$ be a sequence of integers such that $$ \limsup \frac{n_k}{k} = \infty. $$ Is $$ \sum_{k=1}^{\infty} \frac{1}{2^{n_k}} $$ transcendental? sorry line 44 source
Source documentation
Let $n_1 < n_2 < \cdots$ be a sequence of integers such that $$ \limsup \frac{n_k}{k} = \infty. $$ Is $$ \sum_{k=1}^{\infty} \frac{1}{2^{n_k}} $$ transcendental?

depends on sorry what this means
Statement uses: Iff, True, Nat, StrictMono, Eq, EReal, Filter.limsup, HDiv.hDiv, Nat.succ, Filter.atTop, Top.top, Transcendental, Rat, Real, tsum, OfNat.ofNat, HPow.hPow, SummationFilter.unconditional
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
80 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_247 : answer(sorry) ↔ ∀ (n : ℕ → ℕ), (StrictMono n) →
    atTop.limsup (fun k => (n k / k.succ : EReal)) = ⊤ →
    Transcendental ℚ (∑' k, (1 : ℝ) / 2 ^ n k) := by
…
```
theorem Erdos247.erdos_247.variants.strong_condition Erdős proved the answer is yes under the stronger condition that $\limsup \frac{n_k}{k^t} = \infty$ for all $t\geq 1$ sorry line 58 source
Source documentation
Erdős proved the answer is yes under the stronger condition that $\limsup \frac{n_k}{k^t} = \infty$ for all $t\geq 1$.
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).

depends on sorry what this means
Statement uses: Nat, StrictMono, Real, GE.ge, OfNat.ofNat, Eq, EReal, Filter.limsup, HDiv.hDiv, HPow.hPow, Nat.succ, Filter.atTop, Top.top, Transcendental, Rat, tsum, SummationFilter.unconditional
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
78 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_247.variants.strong_condition (n : ℕ → ℕ)
    (hn : StrictMono n)
    (h : ∀ t ≥ (1 : ℝ),
      atTop.limsup (fun k => n k / (k.succ : ℝ) ^ t |>.toEReal) = ⊤) :
    Transcendental ℚ (∑' k, (1 : ℝ) / 2 ^ n k) := by
…
```