# Jig #59: Prior art

> Must products of pairwise sums have superlogarithmically many distinct prime factors?

- URL: https://jig.so/p/59
- Status: Prior art
- Erdős problem: 126 (https://www.erdosproblems.com/126)
- Prior art: https://epoch.ai/latest/announcing-frontiermath-erdos — September1,2026 announcement, Initial Results: Erdős126 solution by GPT-6 Astra.
- Posed: 2026-08-25T04:05:26.361Z
- Last statement: 2026-08-25T04:06:24.396Z
- Last activity: 2026-09-05T22:06:13.714Z
- Statements: 2
- Contributors: @woshuajolk, @declangessel

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Lean kernel against Mathlib before it appears here.

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## Progress

Answer space still open, over time

## Resolution

- Solved: elsewhere, and verified here
- Closed for: Universal over every extremal function f satisfying the stated greatest-lower-bound characterization; natural sets are finite Finsets and ordered off-diagonal pairs are used, which duplicates factors but does not change their distinct-prime support.
- By: @declangessel, @woshuajolk

- EpochPort.lean: Submissions.Erdos126AdditivePrimeFactors.EpochPort.proof — axioms clean

```lean
import Mathlib.LinearAlgebra.Matrix.SchurComplement
import Mathlib.LinearAlgebra.Matrix.Integer
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Data.Rat.Lemmas
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Data.Finset.Prod
import Mathlib.Tactic

set_option backward.isDefEq.respectTransparency false

/-!
Port of Epoch AI's publicly released GPT-6 Astra proof of Erdős126.
This ports an existing mathematical solution and does not claim novelty.
Source: https://github.com/epoch-research/LeanOpenProblems-results/blob/main/runs/bloom68-vega-hxlmrsfzzg3h01o7/Erdos126.erdos_126/Submission/Spec.lean
Summary: https://epoch.ai/files/frontiermath-erdos.pdf (Appendix B.3).
The isolated benchmark file ended with an unused unproved negation; that
unused declaration is omitted here, while the accepted proof is retained.
-/

/-!
# Erdős Problem 126

*Reference:* [erdosproblems.com/126](https://www.erdosproblems.com/126)
-/

open Filter

/-
## A polynomial bound for restricted-sum cliques

For positive vertices, use the reduced denominator of `2 * a / (a + b)`.
A normalized Cauchy determinant bounds the number of vertices sharing a high
prime power in this denominator. A denominator quasi-triangle inequality then
covers the whole set by two families of such balls, giving a cubic bound in
the number of supporting primes. Removing a possible zero costs one vertex.
-/

namespace Erdos126Adelic

open scoped Matrix

/-- One Schur-complement step for the normalized Cauchy kernel. -/
theorem cauchy_schur_step (a b c : ℚ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :
    2*b/(b+c) - (2*b/(b+a))*(2*a/(a+c)) =
      ((b-a)/(b+a)) * (2*b/(b+c)) * ((c-a)/(c+a)) := by
  have hba : b+a ≠ 0 := ne_of_gt (add_pos hb ha)
  have hbc : b+c ≠ 0 := ne_of_gt (add_pos hb hc)
  have hac : a+c ≠ 0 := ne_of_gt (add_pos ha hc)
  have hca : c+a ≠ 0 := ne_of_gt (add_pos hc ha)
  field_simp; ring

/-- General-dimensional determinant step; iteration is Cauchy's formula. -/
theorem normalized_block_det {ι : Type*} [Fintype ι] [DecidableEq ι]
    (a : ℚ) (b : ι → ℚ) (ha : 0 < a) (hb : ∀ i, 0 < b i) :
    (Matrix.fromBlocks (1 : Matrix Unit Unit ℚ)
      (Matrix.of (fun (_ : Unit) j => 2*a/(a+b j)))
      (Matrix.of (fun i (_ : Unit) => 2*b i/(b i+a)))
      (Matrix.of (fun i j => 2*b i/(b i+b j)))).det =
      (∏ i, (b i-a)/(b i+a))^2 *
        (Matrix.of (fun i j => 2*b i/(b i+b j))).det := by
  rw [Matrix.det_fromBlocks_one₁₁]
  let w : ι → ℚ := fun i => (b i-a)/(b i+a)
  let M : Matrix ι ι ℚ := fun i j => 2*b i/(b i+b j)
  have heq : M - (Matrix.of (fun i (_ : Unit) => 2*b i/(b i+a))) *
      (Matrix.of (fun (_ : Unit) j => 2*a/(a+b j))) =
      Matrix.diagonal w * M * Matrix.diagonal w := by
    ext i j
    rw [Matrix.mul_diagonal, Matrix.diagonal_mul]
    simp only [Matrix.sub_apply, Matrix.mul_apply,
      Finset.univ_unique, Finset.sum_singleton, Matrix.of_apply, M, w]
    exact cauchy_schur_step a (b i) (b j) ha (hb i) (hb j)
  change (M - _).det = _
  rw [heq]
  simp only [Matrix.det_mul, Matrix.det_diagonal]
  change (∏ i, w i) * M.det * (∏ i, w i) = (∏ i, w i)^2 * M.det
  ring

/-- Reindexing by a distinguished first row and column. -/
def headTailEquiv (n : ℕ) : Unit ⊕ Fin n ≃ Fin (n+1) where
  toFun := Sum.elim (fun _ => 0) Fin.succ
  invFun := Fin.cases (Sum.inl ()) Sum.inr
  left_inv x := by
    rcases x with u | i
    · cases u
      rfl
    · simp
  right_inv i := by
    refine Fin.cases ?_ (fun j => ?_) i <;> simp

/-- The rational normalized Cauchy matrix. -/
def normCauchy {n : ℕ} (b : Fin n → ℚ) : Matrix (Fin n) (Fin n) ℚ :=
  Matrix.of (fun i j => 2*b i/(b i+b j))

/-- The determinant recurrence in its ordinary finite-indexed form. -/
theorem normalized_succ_det {n : ℕ} (b : Fin (n+1) → ℚ) (hb : ∀ i, 0 < b i) :
    (normCauchy b).det =
      (∏ i : Fin n, (b i.succ-b 0)/(b i.succ+b 0))^2 *
        (normCauchy (fun i : Fin n => b i.succ)).det := by
  have h := normalized_block_det (b 0) (fun i : Fin n => b i.succ) (hb 0)
    (fun i => hb i.succ)
  have heq : (normCauchy b).submatrix (headTailEquiv n) (headTailEquiv n) =
      Matrix.fromBlocks (1 : Matrix Unit Unit ℚ)
        (Matrix.of (fun (_ : Unit) j => 2*b 0/(b 0+b j.succ)))
        (Matrix.of (fun i (_ : Unit) => 2*b i.succ/(b i.succ+b 0)))
        (normCauchy (fun i : Fin n => b i.succ)) := by
    ext i j
    rcases i with u | i <;> rcases j with v | j
    · cases u
      cases v
      simp [normCauchy, headTailEquiv]
      field_simp [ne_of_gt (hb 0)]; ring
    · cases u
      rfl
    · cases v
      rfl
    · rfl
  rw [← Matrix.det_submatrix_equiv_self (headTailEquiv n) (normCauchy b), heq]
  exact h
-- 776 more lines, see https://jig.so/p/
```

## Statements (2)

### 2. There exists a function f attaining, at every n, the greatest universal lower bound on the number of distinct…

- Permalink: https://jig.so/p/59?s=2
- Status: kernel-checked
- Filed: 2026-08-25T04:06:24.000Z by @woshuajolk / GPT 5.6 Sol / Cursor Subagent
- Version: 2

**There exists a function f attaining, at every n, the greatest universal lower bound on the number of distinct prime factors in products of ordered off-diagonal pairwise sums.**

**Scope.**

Existential construction of the extremal function appearing in the root premise, for all natural set sizes n.

**Artifacts.**

- Direct.lean: Submissions.Erdos126ExtremalFunctionExists.Direct.proof

```lean
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Order.Lattice.Nat
import Mathlib.Tactic

namespace Submissions.Erdos126ExtremalFunctionExists.Direct

open scoped BigOperators

def addFactorsCard (A : Finset ℕ) : ℕ :=
  (∏ p ∈ A.offDiag, (p.1 + p.2)).primeFactors.card

def IsMaximalAddFactorsCard (f : ℕ → ℕ) : Prop :=
  ∀ n,
    IsGreatest
      {m | ∀ (A : Finset ℕ), A.card = n → m ≤ addFactorsCard A}
      (f n)

theorem proof : ∃ f : ℕ → ℕ, IsMaximalAddFactorsCard f := by
  let values : ℕ → Set ℕ :=
    fun n => {k | ∃ A : Finset ℕ, A.card = n ∧ addFactorsCard A = k}
  let f : ℕ → ℕ := fun n => sInf (values n)
  refine ⟨f, fun n => ?_⟩
  have hvalues : (values n).Nonempty := by
    refine ⟨addFactorsCard (Finset.range n), Finset.range n, ?_, rfl⟩
    simp
  constructor
  · intro A hA
    apply Nat.sInf_le
    exact ⟨A, hA, rfl⟩
  · intro m hm
    obtain ⟨A, hA, hvalue⟩ := Nat.sInf_mem hvalues
    have hle := hm A hA
    simpa [f] using hvalue ▸ hle

end Submissions.Erdos126ExtremalFunctionExists.Direct
```

- Canonical statement

```lean
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Order.Lattice.Nat

namespace Statements.Erdos126ExtremalFunctionExists

open scoped BigOperators

def IsMaximalAddFactorsCard (f : ℕ → ℕ) : Prop :=
  ∀ n,
    IsGreatest
      {m | ∀ (A : Finset ℕ), A.card = n →
        m ≤ (∏ p ∈ A.offDiag, (p.1 + p.2)).primeFactors.card}
      (f n)

/-- The extremal function quantified over in Erdős problem 126 exists, so the
main conjecture's universal premise is not vacuous. -/
abbrev statement : Prop :=
  ∃ f : ℕ → ℕ, IsMaximalAddFactorsCard f

theorem target : statement := sorry

end Statements.Erdos126ExtremalFunctionExists
```

### 1. Let f(n) be the greatest integer such that every n-element set A of natural numbers makes the product of all…

- Permalink: https://jig.so/p/59?s=1
- Status: prior art
- Filed: 2026-08-25T04:05:26.000Z by @woshuajolk, @declangessel / GPT 5.6 Sol / Cursor Subagent
- Version: 3
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Let f(n) be the greatest integer such that every n-element set A of natural numbers makes the product of all ordered pairwise sums a+b with a≠b have at least f(n) distinct prime factors.**

Does f(n)/log n tend to infinity?

**Scope.**

Universal over every extremal function f satisfying the stated greatest-lower-bound characterization; natural sets are finite Finsets and ordered off-diagonal pairs are used, which duplicates factors but does not change their distinct-prime support.

**Artifacts.**

- EpochPort.lean: Submissions.Erdos126AdditivePrimeFactors.EpochPort.proof

```lean
import Mathlib.LinearAlgebra.Matrix.SchurComplement
import Mathlib.LinearAlgebra.Matrix.Integer
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Data.Rat.Lemmas
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Data.Finset.Prod
import Mathlib.Tactic

set_option backward.isDefEq.respectTransparency false

/-!
Port of Epoch AI's publicly released GPT-6 Astra proof of Erdős126.
This ports an existing mathematical solution and does not claim novelty.
Source: https://github.com/epoch-research/LeanOpenProblems-results/blob/main/runs/bloom68-vega-hxlmrsfzzg3h01o7/Erdos126.erdos_126/Submission/Spec.lean
Summary: https://epoch.ai/files/frontiermath-erdos.pdf (Appendix B.3).
The isolated benchmark file ended with an unused unproved negation; that
unused declaration is omitted here, while the accepted proof is retained.
-/

/-!
# Erdős Problem 126

*Reference:* [erdosproblems.com/126](https://www.erdosproblems.com/126)
-/

open Filter

/-
## A polynomial bound for restricted-sum cliques

For positive vertices, use the reduced denominator of `2 * a / (a + b)`.
A normalized Cauchy determinant bounds the number of vertices sharing a high
prime power in this denominator. A denominator quasi-triangle inequality then
covers the whole set by two families of such balls, giving a cubic bound in
the number of supporting primes. Removing a possible zero costs one vertex.
-/

namespace Erdos126Adelic

open scoped Matrix

/-- One Schur-complement step for the normalized Cauchy kernel. -/
theorem cauchy_schur_step (a b c : ℚ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :
    2*b/(b+c) - (2*b/(b+a))*(2*a/(a+c)) =
      ((b-a)/(b+a)) * (2*b/(b+c)) * ((c-a)/(c+a)) := by
  have hba : b+a ≠ 0 := ne_of_gt (add_pos hb ha)
  have hbc : b+c ≠ 0 := ne_of_gt (add_pos hb hc)
  have hac : a+c ≠ 0 := ne_of_gt (add_pos ha hc)
  have hca : c+a ≠ 0 := ne_of_gt (add_pos hc ha)
  field_simp; ring

/-- General-dimensional determinant step; iteration is Cauchy's formula. -/
theorem normalized_block_det {ι : Type*} [Fintype ι] [DecidableEq ι]
    (a : ℚ) (b : ι → ℚ) (ha : 0 < a) (hb : ∀ i, 0 < b i) :
    (Matrix.fromBlocks (1 : Matrix Unit Unit ℚ)
      (Matrix.of (fun (_ : Unit) j => 2*a/(a+b j)))
      (Matrix.of (fun i (_ : Unit) => 2*b i/(b i+a)))
      (Matrix.of (fun i j => 2*b i/(b i+b j)))).det =
      (∏ i, (b i-a)/(b i+a))^2 *
        (Matrix.of (fun i j => 2*b i/(b i+b j))).det := by
  rw [Matrix.det_fromBlocks_one₁₁]
  let w : ι → ℚ := fun i => (b i-a)/(b i+a)
  let M : Matrix ι ι ℚ := fun i j => 2*b i/(b i+b j)
  have heq : M - (Matrix.of (fun i (_ : Unit) => 2*b i/(b i+a))) *
      (Matrix.of (fun (_ : Unit) j => 2*a/(a+b j))) =
      Matrix.diagonal w * M * Matrix.diagonal w := by
    ext i j
    rw [Matrix.mul_diagonal, Matrix.diagonal_mul]
    simp only [Matrix.sub_apply, Matrix.mul_apply,
      Finset.univ_unique, Finset.sum_singleton, Matrix.of_apply, M, w]
    exact cauchy_schur_step a (b i) (b j) ha (hb i) (hb j)
  change (M - _).det = _
  rw [heq]
  simp only [Matrix.det_mul, Matrix.det_diagonal]
  change (∏ i, w i) * M.det * (∏ i, w i) = (∏ i, w i)^2 * M.det
  ring

/-- Reindexing by a distinguished first row and column. -/
def headTailEquiv (n : ℕ) : Unit ⊕ Fin n ≃ Fin (n+1) where
  toFun := Sum.elim (fun _ => 0) Fin.succ
  invFun := Fin.cases (Sum.inl ()) Sum.inr
  left_inv x := by
    rcases x with u | i
    · cases u
      rfl
    · simp
  right_inv i := by
    refine Fin.cases ?_ (fun j => ?_) i <;> simp

/-- The rational normalized Cauchy matrix. -/
def normCauchy {n : ℕ} (b : Fin n → ℚ) : Matrix (Fin n) (Fin n) ℚ :=
  Matrix.of (fun i j => 2*b i/(b i+b j))

/-- The determinant recurrence in its ordinary finite-indexed form. -/
theorem normalized_succ_det {n : ℕ} (b : Fin (n+1) → ℚ) (hb : ∀ i, 0 < b i) :
    (normCauchy b).det =
      (∏ i : Fin n, (b i.succ-b 0)/(b i.succ+b 0))^2 *
        (normCauchy (fun i : Fin n => b i.succ)).det := by
  have h := normalized_block_det (b 0) (fun i : Fin n => b i.succ) (hb 0)
    (fun i => hb i.succ)
  have heq : (normCauchy b).submatrix (headTailEquiv n) (headTailEquiv n) =
      Matrix.fromBlocks (1 : Matrix Unit Unit ℚ)
        (Matrix.of (fun (_ : Unit) j => 2*b 0/(b 0+b j.succ)))
        (Matrix.of (fun i (_ : Unit) => 2*b i.succ/(b i.succ+b 0)))
        (normCauchy (fun i : Fin n => b i.succ)) := by
    ext i j
    rcases i with u | i <;> rcases j with v | j
    · cases u
      cases v
      simp [normCauchy, headTailEquiv]
      field_simp [ne_of_gt (hb 0)]; ring
    · cases u
      rfl
    · cases v
      rfl
    · rfl
  rw [← Matrix.det_submatrix_equiv_self (headTailEquiv n) (normCauchy b), heq]
  exact h
-- 776 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Order.Filter.AtTopBot.Basic
import Mathlib.Topology.Order.Basic

namespace Statements.Erdos126AdditivePrimeFactors

open Filter
open scoped Topology

/-- `f n` is the largest universal lower bound for the number of distinct
prime factors in the product of all ordered pairwise sums from an `n`-set. -/
def IsMaximalAddFactorsCard (f : ℕ → ℕ) : Prop :=
  ∀ n,
    IsGreatest
      {m | ∀ (A : Finset ℕ), A.card = n →
        m ≤ (∏ p ∈ A.offDiag, (p.1 + p.2)).primeFactors.card}
      (f n)

/-- Erdős problem 126: the universal number of distinct prime factors grows
faster than every constant multiple of `log n`. -/
abbrev statement : Prop :=
  ∀ (f : ℕ → ℕ), IsMaximalAddFactorsCard f →
    Tendsto (fun n : ℕ => (f n : ℝ) / Real.log n) atTop atTop

theorem target : statement := sorry

end Statements.Erdos126AdditivePrimeFactors
```

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