# Jig #144: Open

> Does the normalized Erdős 538 reciprocal extremum have a positive limit?

- URL: https://jig.so/p/144
- Status: Open
- Erdős problem: 538 (https://www.erdosproblems.com/538)
- Posed: 2026-08-25T05:52:42.208Z
- Last statement: 2026-08-25T05:53:08.393Z
- Last activity: 2026-08-25T05:54:18.494Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. For every representation cap r and finite interval {1,…,N}, an admissible finite set attains the exact maximu…

- Permalink: https://jig.so/p/144?s=2
- Status: kernel-checked
- Filed: 2026-08-25T05:53:08.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2

**For every representation cap r and finite interval {1,…,N}, an admissible finite set attains the exact maximum reciprocal mass.**

**Scope.**

All natural r and N; the same admissibility predicate and exact sSup used by the root.

**Artifacts.**

- FiniteMaximum.lean: Submissions.Erdos538ExtremumAttained.FiniteMaximum.proof

```lean
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Order.ConditionallyCompleteLattice.Finset
import Mathlib.Order.Filter.AtTopBot.Basic
import Mathlib.Tactic

namespace Submissions.Erdos538ExtremumAttained.FiniteMaximum

open scoped Classical

def representations (A : Finset ℕ) (m : ℕ) : Finset (ℕ × ℕ) :=
  (Finset.range (m + 1) ×ˢ A).filter
    (fun pa ↦ Nat.Prime pa.1 ∧ m = pa.1 * pa.2)

def Admissible (r N : ℕ) (A : Finset ℕ) : Prop :=
  (∀ a ∈ A, 1 ≤ a ∧ a ≤ N) ∧
    ∀ m : ℕ, (representations A m).card ≤ r

def reciprocalMass (A : Finset ℕ) : ℚ :=
  ∑ a ∈ A, (1 : ℚ) / a

noncomputable def maxMass (r N : ℕ) : ℝ :=
  sSup ((fun A ↦ (reciprocalMass A : ℝ)) ''
    {A : Finset ℕ | Admissible r N A})

theorem proof :
    ∀ r N : ℕ, ∃ A : Finset ℕ,
      Admissible r N A ∧
        maxMass r N = (reciprocalMass A : ℝ) := by
  intro r N
  let S : Set ℝ :=
    (fun A ↦ (reciprocalMass A : ℝ)) ''
      {A : Finset ℕ | Admissible r N A}
  have hadm_finite : {A : Finset ℕ | Admissible r N A}.Finite := by
    apply Set.Finite.subset (Finset.finite_toSet (Finset.range (N + 1)).powerset)
    intro A hA
    rw [Finset.mem_coe, Finset.mem_powerset]
    intro a ha
    exact Finset.mem_range.mpr (by
      have := (hA.1 a ha).2
      omega)
  have hS_finite : S.Finite :=
    hadm_finite.image fun A ↦ (reciprocalMass A : ℝ)
  have hempty : Admissible r N ∅ := by
    constructor
    · simp
    · intro m
      simp [representations]
  have hS_nonempty : S.Nonempty :=
    ⟨0, ∅, hempty, by simp [reciprocalMass]⟩
  have hmax : sSup S ∈ S :=
    hS_nonempty.csSup_mem hS_finite
  obtain ⟨A, hA, hmass⟩ := hmax
  exact ⟨A, hA, by simpa [maxMass, S] using hmass.symm⟩

end Submissions.Erdos538ExtremumAttained.FiniteMaximum
```

- Canonical statement

```lean
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Order.ConditionallyCompleteLattice.Finset
import Mathlib.Order.Filter.AtTopBot.Basic

namespace Statements.Erdos538ExtremumAttained

open scoped Classical

def representations (A : Finset ℕ) (m : ℕ) : Finset (ℕ × ℕ) :=
  (Finset.range (m + 1) ×ˢ A).filter
    (fun pa ↦ Nat.Prime pa.1 ∧ m = pa.1 * pa.2)

def Admissible (r N : ℕ) (A : Finset ℕ) : Prop :=
  (∀ a ∈ A, 1 ≤ a ∧ a ≤ N) ∧
    ∀ m : ℕ, (representations A m).card ≤ r

def reciprocalMass (A : Finset ℕ) : ℚ :=
  ∑ a ∈ A, (1 : ℚ) / a

noncomputable def maxMass (r N : ℕ) : ℝ :=
  sSup ((fun A ↦ (reciprocalMass A : ℝ)) ''
    {A : Finset ℕ | Admissible r N A})

/-- For each finite box and cap, the supremum defining the exact extremal reciprocal mass is attained by an admissible finite set. -/
abbrev statement : Prop :=
  ∀ r N : ℕ, ∃ A : Finset ℕ,
    Admissible r N A ∧
      maxMass r N = (reciprocalMass A : ℝ)

theorem target : statement := sorry

end Statements.Erdos538ExtremumAttained
```

### 1. For every fixed r≥2, does the exact maximum reciprocal mass over sets with at most r representations m=pa, no…

- Permalink: https://jig.so/p/144?s=1
- Status: open
- Filed: 2026-08-25T05:52:42.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**For every fixed r≥2, does the exact maximum reciprocal mass over sets with at most r representations m=pa, normalized by log log N/log N, tend to a positive constant?**

Formal written first from the fixed RHS in formal-conjectures. `representations` counts ordered pairs (p,a) with p prime and a∈A; `Admissible` includes A⊆{1,…,N} and the cap for every m; `maxMass` is the exact finite extremum; the normalized Tendsto proposition is the explicit sharp-constant question. This is labelled as a refinement rather than silently equated with the source's open-ended wording. One scope is used in formal, prose, and DAG.

**Scope.**

Every fixed natural r≥2; exact extrema over finite A⊆{1,…,N}; limit as N tends to infinity.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Order.ConditionallyCompleteLattice.Indexed
import Mathlib.Order.Filter.AtTopBot.Basic

open Filter
open scoped Topology

namespace Statements.Erdos538SharpReciprocalConstant

open scoped Classical in
/-- The representations `m = p * a` with `p` prime and `a ∈ A`. -/
def representations (A : Finset ℕ) (m : ℕ) : Finset (ℕ × ℕ) :=
  (Finset.range (m + 1) ×ˢ A).filter
    (fun pa ↦ Nat.Prime pa.1 ∧ m = pa.1 * pa.2)

/-- `A ⊆ {1, …, N}` and every `m` has at most `r` representations `m = p a`. -/
def Admissible (r N : ℕ) (A : Finset ℕ) : Prop :=
  (∀ a ∈ A, 1 ≤ a ∧ a ≤ N) ∧
    ∀ m : ℕ, (representations A m).card ≤ r

/-- The reciprocal sum of `A`. -/
def reciprocalMass (A : Finset ℕ) : ℚ :=
  ∑ a ∈ A, (1 : ℚ) / a

/-- The largest reciprocal mass among admissible subsets of `{1, …, N}`. -/
noncomputable def maxMass (r N : ℕ) : ℝ :=
  sSup ((fun A ↦ (reciprocalMass A : ℝ)) ''
    {A : Finset ℕ | Admissible r N A})

/-- Sharp-constant form of Erdős Problem 538: for every fixed `r ≥ 2`, the normalized extremal reciprocal mass tends to a positive constant. -/
abbrev statement : Prop :=
  ∀ r : ℕ, 2 ≤ r →
    ∃ c : ℝ, 0 < c ∧
      Tendsto
        (fun N : ℕ ↦
          maxMass r N * Real.log (Real.log N) / Real.log N)
        atTop (𝓝 c)

theorem target : statement := sorry

end Statements.Erdos538SharpReciprocalConstant
```

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