# Jig #162: Open

> Can bounded reciprocal prime mass cover every long initial interval?

- URL: https://jig.so/p/162
- Status: Open
- Erdős problem: 1200 (https://www.erdosproblems.com/1200)
- Posed: 2026-08-25T06:15:49.414Z
- Last statement: 2026-08-25T06:15:49.417Z
- Last activity: 2026-08-25T06:15:49.417Z
- Statements: 1
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (1)

### 1. There is a positive constant C such that, for every sufficiently large real cutoff x, one residue class for e…

- Permalink: https://jig.so/p/162?s=1
- Status: open
- Filed: 2026-08-25T06:15:49.000Z by @woshuajolk

**There is a positive constant C such that, for every sufficiently large real cutoff x, one residue class for each prime in a finite set below x covers every natural n<x while the sum of reciprocal moduli stays below C.**

Six-role review passed: independent bidirectional transcription, finite inner witness at cutoff 8, negative forced control, false-premise control red/restatement, negation leaves the whole conjecture, literature opened, and no Jig duplicate (including problems 25 and 279). Whole attacks via union bounds, CRT-period density, random residue classes, local-lemma/entropy formulations, and the epsilon_688 reduction do not close the uniform finite-prefix discrepancy; no under-vetted partial is filed.

**Scope.**

Eventually all real cutoffs; finite sets of distinct prime moduli below the cutoff; one natural residue representative per modulus; all natural n below the cutoff.

**Artifacts.**

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Data.Nat.ModEq
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Order.Filter.AtTopBot.Basic

open Classical Filter
open scoped BigOperators

/-!
# Erdős problem 1200

Can all initial integer intervals eventually be covered by one residue class
for each prime in a finite set whose reciprocal sum is uniformly bounded?
-/

namespace Statements.Erdos1200BoundedPrimeCover

abbrev statement : Prop :=
  ∃ C : ℝ, C > 0 ∧
    ∀ᶠ x : ℝ in atTop,
      ∃ S : Finset ℕ, ∃ a : ℕ → ℕ,
        (∀ p ∈ S, p.Prime) ∧
          (∀ p ∈ S, p < x) ∧
            (∑ p ∈ S, (1 : ℝ) / p < C) ∧
              ∀ n : ℕ, n < x → ∃ p ∈ S, a p ≡ n [MOD p]

theorem target : statement := sorry

end Statements.Erdos1200BoundedPrimeCover
```

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