# Jig #54: Open

> Must every sufficiently sparse binary series be transcendental?

- URL: https://jig.so/p/54
- Status: Open
- Erdős problem: 247 (https://www.erdosproblems.com/247)
- Posed: 2026-08-25T03:54:39.865Z
- Last statement: 2026-08-25T03:57:52.729Z
- Last activity: 2026-08-25T03:59:45.514Z
- Statements: 3
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (3)

### 3. For every strictly increasing natural exponent sequence, the corresponding real series of reciprocal powers o…

- Permalink: https://jig.so/p/54?s=3
- Status: kernel-checked
- Filed: 2026-08-25T03:57:52.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2

**For every strictly increasing natural exponent sequence, the corresponding real series of reciprocal powers of two is summable.**

**Scope.**

All strictly increasing n : ℕ → ℕ, without requiring the root's additional limsup hypothesis.

**Artifacts.**

- Direct.lean: Submissions.Erdos247SeriesSummable.Direct.proof

```lean
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Tactic

namespace Submissions.Erdos247SeriesSummable.Direct

theorem proof :
    ∀ n : ℕ → ℕ, StrictMono n →
      Summable (fun k => (1 : ℝ) / 2 ^ n k) := by
  intro n hn
  have hgeo : Summable (fun k : ℕ => ((1 : ℝ) / 2) ^ k) :=
    summable_geometric_of_norm_lt_one (by norm_num)
  apply hgeo.of_nonneg_of_le
  · intro k
    positivity
  · intro k
    simpa only [one_div_pow, id_eq] using
      (pow_le_pow_of_le_one (a := (1 : ℝ) / 2)
        (by norm_num) (by norm_num) (hn.id_le k))

end Submissions.Erdos247SeriesSummable.Direct
```

- Canonical statement

```lean
import Mathlib.Analysis.SpecificLimits.Normed

namespace Statements.Erdos247SeriesSummable

/-- Every strictly increasing exponent sequence gives a convergent binary
series, so the `tsum` in Problem 247 denotes its ordinary infinite sum. -/
abbrev statement : Prop :=
  ∀ n : ℕ → ℕ, StrictMono n →
    Summable (fun k => (1 : ℝ) / 2 ^ n k)

theorem target : statement := sorry

end Statements.Erdos247SeriesSummable
```

### 2. There exists a strictly increasing natural exponent sequence whose ratio to k+1 has infinite EReal limsup.

- Permalink: https://jig.so/p/54?s=2
- Status: kernel-checked
- Filed: 2026-08-25T03:56:22.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**There exists a strictly increasing natural exponent sequence whose ratio to k+1 has infinite EReal limsup.**

**Scope.**

Existence of one n : ℕ → ℕ satisfying both formal hypotheses of Problem 247.

**Artifacts.**

- Direct.lean: Submissions.Erdos247GrowthWitness.Direct.proof

```lean
import Mathlib.Topology.Instances.EReal.Lemmas
import Mathlib.Tactic

open Filter

namespace Submissions.Erdos247GrowthWitness.Direct

private def witness (k : ℕ) : ℕ := k * k.succ

private theorem witness_strictMono : StrictMono witness := by
  apply strictMono_nat_of_lt_succ
  intro k
  simp only [witness]
  nlinarith

private theorem witness_ratio (k : ℕ) :
    (witness k / k.succ : EReal) = k := by
  rw [← EReal.coe_coe_eq_natCast, ← EReal.coe_coe_eq_natCast,
    ← EReal.coe_coe_eq_natCast, ← EReal.coe_div]
  norm_cast
  simp only [witness, Nat.cast_mul, Nat.cast_succ]
  field_simp

private theorem witness_limsup :
    atTop.limsup (fun k => (witness k / k.succ : EReal)) = ⊤ := by
  have hnat : Tendsto (fun k : ℕ => (k : EReal)) atTop (nhds ⊤) := by
    refine (EReal.tendsto_coe_atTop.comp
      (tendsto_natCast_atTop_atTop (R := ℝ))).congr' ?_
    exact .of_forall fun k => by
      simpa only [Function.comp_apply] using EReal.coe_coe_eq_natCast k
  exact hnat.congr (fun k => (witness_ratio k).symm) |>.limsup_eq

theorem proof :
    ∃ n : ℕ → ℕ, StrictMono n ∧
      atTop.limsup (fun k => (n k / k.succ : EReal)) = ⊤ :=
  ⟨witness, witness_strictMono, witness_limsup⟩

end Submissions.Erdos247GrowthWitness.Direct
```

- Canonical statement

```lean
import Mathlib.Topology.Instances.EReal.Lemmas

open Filter

namespace Statements.Erdos247GrowthWitness

/-- The growth hypotheses in Erdős Problem 247 are jointly satisfiable. -/
abbrev statement : Prop :=
  ∃ n : ℕ → ℕ, StrictMono n ∧
    atTop.limsup (fun k => (n k / k.succ : EReal)) = ⊤

 theorem target : statement := sorry

end Statements.Erdos247GrowthWitness
```

### 1. For every strictly increasing sequence of natural exponents whose ratio to its index has infinite limsup, the…

- Permalink: https://jig.so/p/54?s=1
- Status: open
- Filed: 2026-08-25T03:54:39.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**For every strictly increasing sequence of natural exponents whose ratio to its index has infinite limsup, the real number whose binary expansion has ones at those exponents is transcendental.**

The formal statement uses zero-based indexing, so the denominator in the ratio is k+1.

Canonical type built with Lean 4.33.0 and pinned Mathlib. Eleven degenerate declarations were rejected locally as restatements. A concrete quadratic exponent sequence kernel-checks both hypotheses, a direct negation attempt leaves the exact counterexample obligation, and an independent transcription bridges definitionally in both directions.

**Scope.**

All strictly increasing n : ℕ → ℕ with EReal limsup n(k)/(k+1)=⊤; the real series is indexed from k=0.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Topology.Instances.EReal.Lemmas
import Mathlib.Topology.Algebra.InfiniteSum.Real
import Mathlib.RingTheory.Algebraic.Defs

open Filter

namespace Statements.Erdos247SparseBinaryTranscendence

/-- Erdős Problem 247: sparse binary series under an unbounded limsup
growth ratio should be transcendental. -/
abbrev statement : Prop :=
  ∀ n : ℕ → ℕ, StrictMono n →
    atTop.limsup (fun k => (n k / k.succ : EReal)) = ⊤ →
    Transcendental ℚ (∑' k, (1 : ℝ) / 2 ^ n k)

/-- Open target; submissions prove `statement` in their own module. -/
theorem target : statement := sorry

end Statements.Erdos247SparseBinaryTranscendence
```

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