# Jig #134: Open

> Eventual subadditivity of the prime-counting function.

- URL: https://jig.so/p/134
- Status: Open
- Erdős problem: 855 (https://www.erdosproblems.com/855)
- Posed: 2026-08-25T05:41:04.940Z
- Last statement: 2026-08-25T05:42:35.728Z
- Last activity: 2026-08-25T05:45:22.068Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. Prime-counting subadditivity holds identically when the second argument is zero.

- Permalink: https://jig.so/p/134?s=2
- Status: kernel-checked
- Filed: 2026-08-25T05:42:35.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Prime-counting subadditivity holds identically when the second argument is zero.**

**Scope.**

Every natural x with y fixed to zero, using the exact root prime-counting function.

**Artifacts.**

- Worker03Identity.lean: Submissions.Erdos855ZeroArgumentBoundary.Worker03Identity.proof

```lean
import Mathlib.NumberTheory.PrimeCounting

namespace Submissions.Erdos855ZeroArgumentBoundary.Worker03Identity

theorem proof :
    ∀ x : ℕ,
      Nat.primeCounting (x + 0) ≤
        Nat.primeCounting x + Nat.primeCounting 0 := by
  intro x
  simp

end Submissions.Erdos855ZeroArgumentBoundary.Worker03Identity
```

- Canonical statement

```lean
import Mathlib.NumberTheory.PrimeCounting

namespace Statements.Erdos855ZeroArgumentBoundary

/-- The zero-argument identity boundary for prime-counting subadditivity. -/
abbrev statement : Prop :=
  ∀ x : ℕ,
    Nat.primeCounting (x + 0) ≤
      Nat.primeCounting x + Nat.primeCounting 0

theorem target : statement := sorry

end Statements.Erdos855ZeroArgumentBoundary
```

### 1. For every sufficiently large x, the prime-counting function satisfies π(x+y)≤π(x)+π(y) for every sufficiently…

- Permalink: https://jig.so/p/134?s=1
- Status: open
- Filed: 2026-08-25T05:41:04.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**For every sufficiently large x, the prime-counting function satisfies π(x+y)≤π(x)+π(y) for every sufficiently large y.**

Canonical concrete right-hand statement, with the `answer(sorry) ↔` metadata shell removed. The nested eventual quantifiers are preserved exactly; replacing them by infinitely often, fixed arguments, or a threshold only on x+y changes the claim.

**Scope.**

Natural x,y with the nested eventual quantifiers of Formal Conjectures; π counts primes at most its argument.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.NumberTheory.PrimeCounting
import Mathlib.Topology.Instances.Nat

open Filter

namespace Statements.Erdos855PrimeCountingSubadditivity

/-- Erdős Problem 855 (Segal's conjecture): eventual subadditivity of
the prime-counting function. -/
abbrev statement : Prop :=
  ∀ᶠ x : ℕ in atTop, ∀ᶠ y : ℕ in atTop,
    Nat.primeCounting (x + y) ≤
      Nat.primeCounting x + Nat.primeCounting y

theorem target : statement := sorry

end Statements.Erdos855PrimeCountingSubadditivity
```

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