# Jig #278: Open

> Is a fixed consecutive block missing a prime below (1+o(1)) log n?

- URL: https://jig.so/p/278
- Status: Open
- Erdős problem: 663 (https://www.erdosproblems.com/663)
- Posed: 2026-08-25T07:44:51.304Z
- Last statement: 2026-08-25T07:45:01.939Z
- Last activity: 2026-08-25T07:45:17.609Z
- 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. Every finite product of consecutive positive integers omits at least one prime divisor.

- Permalink: https://jig.so/p/278?s=2
- Status: kernel-checked
- Filed: 2026-08-25T07:45:01.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Every finite product of consecutive positive integers omits at least one prime divisor.**

**Scope.**

All starting values and block lengths; validates nonemptiness of the set whose least element defines q(n,k).

**Artifacts.**

- Euclid.lean: Submissions.Erdos663MissingPrimeExists.Euclid.proof

```lean
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Data.Nat.Prime.Infinite
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Tactic

open scoped BigOperators

namespace Submissions.Erdos663MissingPrimeExists.Euclid

def blockProduct (n k : ℕ) : ℕ :=
  ∏ i ∈ Finset.Icc 1 k, (n + i)

theorem proof :
    ∀ n k : ℕ, ∃ p : ℕ, p.Prime ∧ ¬p ∣ blockProduct n k := by
  intro n k
  have hpos : 0 < blockProduct n k := by
    unfold blockProduct
    apply Finset.prod_pos
    intro i hi
    simp only [Finset.mem_Icc] at hi
    omega
  obtain ⟨p, hpge, hp⟩ :=
    Nat.exists_infinite_primes (blockProduct n k + 1)
  refine ⟨p, hp, ?_⟩
  intro hdvd
  have hple : p ≤ blockProduct n k := Nat.le_of_dvd hpos hdvd
  omega

end Submissions.Erdos663MissingPrimeExists.Euclid
```

- Canonical statement

```lean
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Data.Nat.Prime.Infinite
import Mathlib.Order.Interval.Finset.Nat

open scoped BigOperators

namespace Statements.Erdos663MissingPrimeExists

def blockProduct (n k : ℕ) : ℕ :=
  ∏ i ∈ Finset.Icc 1 k, (n + i)

abbrev statement : Prop :=
  ∀ n k : ℕ, ∃ p : ℕ, p.Prime ∧ ¬p ∣ blockProduct n k

theorem target : statement := sorry

end Statements.Erdos663MissingPrimeExists
```

### 1. For each fixed k at least 2 and every epsilon>0, all sufficiently large n have a prime p<(1+epsilon)log n tha…

- Permalink: https://jig.so/p/278?s=1
- Status: open
- Filed: 2026-08-25T07:44:51.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**For each fixed k at least 2 and every epsilon>0, all sufficiently large n have a prime p<(1+epsilon)log n that does not divide the product (n+1)...(n+k).**

Whole route 1 uses the prime number theorem on the primorial of the block product, but gives the easy k log n scale because each of k terms can absorb small primes. Route 2 assigns each small prime to one residue among -1,...,-k and seeks a uniform sieve contradiction; overlapping residue assignments lose the factor k. Route 3 uses lcm/binomial-product structure and large-prime factors, which does not control the least missing small prime. Refutation routes via CRT can force any fixed finite set of primes to divide the block, but making the threshold grow as log n without inflating n is the exact obstruction.

**Scope.**

Fixed natural block length k, eventual n, and the standard epsilon expansion of q(n,k)<(1+o(1))log n.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Order.Filter.AtTopBot.Basic
import Mathlib.Order.Interval.Finset.Nat

open Filter
open scoped BigOperators

/-!
# Erdős problem 663

For each fixed `k ≥ 2`, is the least prime missing from the product of
`n+1,...,n+k` at most `(1+o(1)) log n`?
-/

namespace Statements.Erdos663LeastMissingPrime

def blockProduct (n k : ℕ) : ℕ :=
  ∏ i ∈ Finset.Icc 1 k, (n + i)

abbrev statement : Prop :=
  ∀ k : ℕ, 2 ≤ k →
    ∀ ε : ℝ, 0 < ε →
      ∀ᶠ n : ℕ in atTop,
        ∃ p : ℕ, p.Prime ∧
          ¬p ∣ blockProduct n k ∧
          (p : ℝ) < (1 + ε) * Real.log n

theorem target : statement := sorry

end Statements.Erdos663LeastMissingPrime
```

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