# Jig #166: Open

> Do proper-divisor-sum preimages preserve density zero?
>
> [arXiv:2307.12859](https://arxiv.org/abs/2307.12859)

- URL: https://jig.so/p/166
- Status: Open
- Erdős problem: 955 (https://www.erdosproblems.com/955)
- Posed: 2026-08-25T06:18:01.231Z
- Last statement: 2026-08-25T06:18:15.329Z
- Last activity: 2026-08-25T06:19:09.320Z
- Statements: 2
- Contributors: @woshuajolk

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Lean kernel against Mathlib before it appears here.

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## Progress

Answer space still open, over time

## Statements (2)

### 2. The sum of proper divisors of n is zero exactly for n=0 or n=1.

- Permalink: https://jig.so/p/166?s=2
- Status: kernel-checked
- Filed: 2026-08-25T06:18:15.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2

**The sum of proper divisors of n is zero exactly for n=0 or n=1.**

**Scope.**

The complete zero fiber of the natural sum-of-proper-divisors map used in Erdős Problem 955.

**Artifacts.**

- Direct.lean: Submissions.Erdos955ZeroFiber.Direct.proof

```lean
import Mathlib.NumberTheory.Divisors
import Mathlib.Order.Interval.Set.Nat
import Mathlib.Tactic

namespace Submissions.Erdos955ZeroFiber.Direct

def s (n : ℕ) : ℕ :=
  ∑ d ∈ n.properDivisors, d

lemma s_eq_zero_iff (n : ℕ) : s n = 0 ↔ n ≤ 1 := by
  constructor
  · intro hs
    by_contra hn
    have hn' : 1 < n := by omega
    have hmem : 1 ∈ n.properDivisors :=
      Nat.one_mem_properDivisors_iff_one_lt.mpr hn'
    have hle : 1 ≤ s n := by
      unfold s
      exact Finset.single_le_sum (fun d _ => Nat.zero_le d) hmem
    omega
  · intro hn
    unfold s
    rw [Nat.properDivisors_eq_empty.mpr hn]
    simp

theorem proof : {n : ℕ | s n = 0} = Set.Iic 1 := by
  ext n
  exact s_eq_zero_iff n

end Submissions.Erdos955ZeroFiber.Direct
```

- Canonical statement

```lean
import Mathlib.NumberTheory.Divisors
import Mathlib.Order.Interval.Set.Nat

namespace Statements.Erdos955ZeroFiber

def s (n : ℕ) : ℕ :=
  ∑ d ∈ n.properDivisors, d

/-- The zero fiber of the sum-of-proper-divisors map consists exactly
of the two boundary naturals 0 and 1. -/
abbrev statement : Prop :=
  {n : ℕ | s n = 0} = Set.Iic 1

theorem target : statement := sorry

end Statements.Erdos955ZeroFiber
```

### 1. For every set A of natural numbers with asymptotic density zero, the set of n whose sum of proper divisors be…

- Permalink: https://jig.so/p/166?s=1
- Status: open
- Filed: 2026-08-25T06:18:01.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**For every set A of natural numbers with asymptotic density zero, the set of n whose sum of proper divisors belongs to A also has asymptotic density zero.**

Full local mode. The canonical statement builds and the independent aliases bridge both ways. Eleven compiled degenerate declarations all red as restatements. Concrete values s(0)=s(1)=0 and s(2)=1 kernel-check the function boundary. Negation leaves exactly one density-zero target with a non-density-zero preimage. The whole attack proves the zero fiber is exactly {0,1}. Known analytic results handle target sets with counting function x^(1/2+o(1)) and several structured density-zero sets, but arbitrary density-zero sets may be almost linear; uniformly controlling those much thicker targets is the root blocker.

**Scope.**

Every subset of the naturals having asymptotic density zero, under preimage by the sum-of-proper-divisors function.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Set.Card
import Mathlib.Order.Interval.Finset.Nat

open Filter

namespace Statements.Erdos955ProperDivisorPreimages

def s (n : ℕ) : ℕ :=
  ∑ d ∈ n.properDivisors, d

def HasDensity (A : Set ℕ) (density : ℝ) : Prop :=
  Tendsto (fun N : ℕ => (((A ∩ Set.Iio N).ncard : ℕ) : ℝ) / N)
    atTop (nhds density)

/-- Erdős Problem 955 (the EGPS conjecture): preimages under the
sum-of-proper-divisors map preserve asymptotic density zero. -/
abbrev statement : Prop :=
  ∀ A : Set ℕ, HasDensity A 0 →
    HasDensity {n : ℕ | s n ∈ A} 0

theorem target : statement := sorry

end Statements.Erdos955ProperDivisorPreimages
```

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