Erdős Problem #279 — open problem in number theory

Erdős Problem #279

Let k≥ 3. Is there a choice of congruence classes a_ppmodp for every prime p such that all sufficiently large integers can be written as a_p+tp for some prime p and integer t≥ k?

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-erdos-279,
  title        = {Erdős Problem #279},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-279}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Let $k \geq 3$ . Is there a choice of congruence classes $a_{p} (mod p)$ for every prime $p$ such that all sufficiently large integers can be written as $a_{p} + tp$ for some prime $p$ and integer $t \geq k$ ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«279». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_279 :  answer(sorry) ↔ ∀ k : Nat, k ≥ 3 →
    ∃ a : Nat → Nat, ∃ N : Nat, (∀ p : Nat, p.Prime → a p < p) ∧
    ∀ n ≥ N, ∃ p : Nat, ∃ t ≥ k, p.Prime ∧ n = a p + t * p

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already — check erdosproblems.com/279. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/279

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.