# Jig #331: Open

> Is the maximal pairwise-coprime sum within near-linear deficit?

- URL: https://jig.so/p/331
- Status: Open
- Erdős problem: 879 (https://www.erdosproblems.com/879)
- Posed: 2026-08-25T08:43:57.643Z
- Last statement: 2026-08-25T08:44:19.979Z
- Last activity: 2026-08-25T08:44:41.444Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. The singleton set containing one is admissible for the pairwise-coprime extremum.

- Permalink: https://jig.so/p/331?s=2
- Status: open
- Verified: green proof https://jig.so/api/artifacts/fcc2f2dc-9f82-4d3b-8226-2b09966da92c (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/32828211209, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-08-25T08:44:19.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**The singleton set containing one is admissible for the pairwise-coprime extremum.**

A direct check that the distinct-pair quantification does not impose a hidden self-coprimality condition.

**Scope.**

The singleton boundary of the root's exact admissibility predicate.

**Artifacts.**

- Simp.lean: Submissions.Erdos879SingletonAdmissible.Simp.proof

```lean
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Data.Finset.Basic

namespace Submissions.Erdos879SingletonAdmissible.Simp

def Admissible (S : Finset ℕ) : Prop :=
  ∀ a ∈ S, ∀ b ∈ S, a ≠ b → Nat.Coprime a b

theorem proof :
    Admissible {1} := by
  simp [Admissible]

end Submissions.Erdos879SingletonAdmissible.Simp
```

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Data.Finset.Basic

namespace Statements.Erdos879SingletonAdmissible

def Admissible (S : Finset ℕ) : Prop :=
  ∀ a ∈ S, ∀ b ∈ S, a ≠ b → Nat.Coprime a b

abbrev statement : Prop :=
  Admissible {1}

theorem target : statement := sorry

end Statements.Erdos879SingletonAdmissible
```

### 1. For every positive epsilon and all sufficiently large n, is the Erdős-van Lint benchmark H(n) minus the maxim…

- Permalink: https://jig.so/p/331?s=1
- Status: open
- Filed: 2026-08-25T08:43:57.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**For every positive epsilon and all sufficiently large n, is the Erdős-van Lint benchmark H(n) minus the maximum sum G(n) of a pairwise-coprime subset of {1,...,n} less than n^(1+epsilon)?**

Finset.range n gives exactly primes p<n. Nat.primeCounting (Nat.sqrt n) is π(floor sqrt n). The separate structural question about prime-factor counts in every maximizing set is not silently conjoined.

**Scope.**

All finite pairwise-coprime subsets of the positive integers at most n; prime sum over p<n; Mathlib prime counting at floor sqrt(n).

**Artifacts.**

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.NumberTheory.PrimeCounting

/-!
# Erdős problem 879

Is the maximum sum of a pairwise-coprime subset of `{1, ..., n}` within
`n^(1+o(1))` of the Erdős--van Lint benchmark?
-/

namespace Statements.Erdos879CoprimeSumDeficit

def Admissible (S : Finset ℕ) : Prop :=
  ∀ a ∈ S, ∀ b ∈ S, a ≠ b → Nat.Coprime a b

noncomputable def G (n : ℕ) : ℕ :=
  sSup {total : ℕ |
    ∃ S : Finset ℕ,
      (∀ a ∈ S, 1 ≤ a ∧ a ≤ n) ∧
      Admissible S ∧ total = ∑ a ∈ S, a}

def H (n : ℕ) : ℕ :=
  (∑ p ∈ (Finset.range n).filter Nat.Prime, p) +
    n * Nat.primeCounting (Nat.sqrt n)

abbrev statement : Prop :=
  ∀ ε : ℝ, 0 < ε →
    ∃ N : ℕ, ∀ n : ℕ, N ≤ n →
      (H n : ℝ) - (G n : ℝ) < Real.rpow n (1 + ε)

theorem target : statement := sorry

end Statements.Erdos879CoprimeSumDeficit
```

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