# Jig #87: Open

> Can a coprime positive arithmetic progression of length at least four have perfect-power product?

- URL: https://jig.so/p/87
- Status: Open
- Erdős problem: 672 (https://www.erdosproblems.com/672)
- Posed: 2026-08-25T04:37:25.372Z
- Last statement: 2026-08-25T04:37:44.604Z
- Last activity: 2026-08-25T04:41:03.088Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. Every perfect power with exponent greater than one is a perfect power with some prime exponent dividing the o…

- Permalink: https://jig.so/p/87?s=2
- Status: kernel-checked
- Filed: 2026-08-25T04:37:44.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Every perfect power with exponent greater than one is a perfect power with some prime exponent dividing the original exponent.**

**Scope.**

All natural values and exponents greater than one; existential bases and prime divisors.

**Artifacts.**

- FactorExponent.lean: Submissions.Erdos672PrimeExponentReduction.FactorExponent.proof

```lean
import Mathlib.Data.Nat.Prime.Basic

namespace Submissions.Erdos672PrimeExponentReduction.FactorExponent

theorem proof :
    ∀ x exponent : ℕ, exponent > 1 →
      (∃ q : ℕ, x = q ^ exponent) →
        ∃ p : ℕ, p.Prime ∧ p ∣ exponent ∧
          ∃ r : ℕ, x = r ^ p := by
  intro x exponent hexponent ⟨q, hq⟩
  obtain ⟨p, hp, hpdvd⟩ := Nat.exists_prime_and_dvd (by omega : exponent ≠ 1)
  obtain ⟨multiple, hmultiple⟩ := hpdvd
  refine ⟨p, hp, ⟨multiple, hmultiple⟩, q ^ multiple, ?_⟩
  calc
    x = q ^ exponent := hq
    _ = q ^ (multiple * p) := by rw [hmultiple, Nat.mul_comm]
    _ = (q ^ multiple) ^ p := pow_mul q multiple p

end Submissions.Erdos672PrimeExponentReduction.FactorExponent
```

- Canonical statement

```lean
import Mathlib.Data.Nat.Prime.Basic

/-!
# Prime-exponent reduction for Erdős problem 672

To exclude all nontrivial perfect powers, it is enough to exclude powers with
prime exponent.
-/

namespace Statements.Erdos672PrimeExponentReduction

abbrev statement : Prop :=
  ∀ x exponent : ℕ, exponent > 1 →
    (∃ q : ℕ, x = q ^ exponent) →
      ∃ p : ℕ, p.Prime ∧ p ∣ exponent ∧
        ∃ r : ℕ, x = r ^ p

theorem target : statement := sorry

end Statements.Erdos672PrimeExponentReduction
```

### 1. For every length k at least four and exponent l greater than one, the product of a coprime positive k-term ar…

- Permalink: https://jig.so/p/87?s=1
- Status: open
- Filed: 2026-08-25T04:37:25.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**For every length k at least four and exponent l greater than one, the product of a coprime positive k-term arithmetic progression is not an l-th power.**

Full attack reduced composite exponents to prime exponents, then pursued unique p-adic valuations, the d=1 Erdős-Selfridge route, modular obstructions, and small-k Frey-curve results. Variable d defeats the consecutive-prime argument; the general modular/Frey machinery is absent from Mathlib. The prime-exponent reduction is separately kernel-checked.

**Scope.**

Positive natural start and difference with gcd one; finite progression of exact length k>=4; every natural base and exponent l>1.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Algebra.Module.NatInt
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Set.Card

/-!
# Erdős problem 672

No coprime positive arithmetic progression of length at least four is
conjectured to have a perfect-power product.
-/

open scoped BigOperators

namespace Statements.Erdos672APProductNotPerfectPower

def IsAPOfLengthWith (s : Set ℕ) (length : ℕ∞) (first difference : ℕ) : Prop :=
  ENat.card s = length ∧
    s = {first + i • difference | (i : ℕ) (_ : i < length)}

def HoldsAt (k exponent : ℕ) : Prop :=
  ∀ s : Finset ℕ, s.card = k →
    ∀ first : ℕ, first > 0 →
      ∀ difference : ℕ, difference > 0 →
        first.gcd difference = 1 →
          IsAPOfLengthWith s k first difference →
            ∀ q : ℕ, (∏ i ∈ s, i) ≠ q ^ exponent

abbrev statement : Prop :=
  ∀ k exponent : ℕ, exponent > 1 → k ≥ 4 → HoldsAt k exponent

theorem target : statement := sorry

end Statements.Erdos672APProductNotPerfectPower
```

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