Erdős (Erdos) Problem #687: status in 2026

Erdős Problem #687

Let Y(x) be the maximal y such that there exists a choice of congruence classes a_p for all primes p≤ x such that every integer in [1,y] is congruent to at least one of the a_ppmodp. Give good estimates for Y(x). We formalise this as determining Y(x) up to constant factors.

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-08-31). Erdős offered a prize of $1000.

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #687 as open (status updated 31 August 2025).

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #687 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: research statements in the file	3 open, 3 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), Formal Conjectures (checked 2026-11-12). How it works.

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.

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Cite

@misc{cairn-erdos-687,
  title        = {Erdős Problem #687},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-687}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

erdos_687.parts.i. Let $Y (x)$ be the maximal $y$ such that there exists a choice of congruence classes $a_{p}$ for all primes $p \leq x$ such that every integer in $[1, y]$ is congruent to at least one of the $a_{p} (mod p)$ . Give good estimates for $Y (x)$ .

We formalise this as determining $Y (x)$ up to constant factors.

erdos_687.parts.iii. Or, even stronger, can one prove that $Y (x) ≪ x^{1 + o (1)}$ ?

We read $Y (x) ≪ x^{1 + o (1)}$ as $Y (x) = O (x^{1 + ε})$ for every fixed $ε > 0$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«687» (2 statements).

theorem erdos_687.parts.i :
    (fun x : ℕ ↦ (Y x : ℝ)) =Θ[atTop] erdos_687.parts.i.answer

theorem erdos_687.parts.iii :
    ∀ ε > (0 : ℝ), (fun x : ℕ ↦ (Y x : ℝ)) =O[atTop] (fun x : ℕ ↦ (x : ℝ) ^ (1 + ε))

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
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/687. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_687.variants.maier_pomerance — Maier and Pomerance [MaPo90] conjectured that Y(x) ≪ x(log x)^2+o(1) (in this form it is stated in [FGKMT18]).

References

erdosproblems.com/687
Ben Green's Open Problem 46: the same question; see Green46 in FormalConjectures.GreensOpenProblems.«46».
OEIS A058989: the values $Y (p_{n})$ , where $p_{n}$ is the $n$ -th prime.
OEIS A048670: Jacobsthal's function of $\prod_{p \leq p_{n}} p$ , which is $Y (p_{n}) + 1$ .
[Er79d] Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. 33 (1979), 71-80.
[Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. 6 (1980), 89-115.
[Er96b] Erdős, Paul, Some problems I presented or planned to present in my short talk. Analytic number theory, Vol. 1 (Allerton Park, IL, 1995), Progr. Math. 138 (1996), 333-335.
[Iw78] Iwaniec, Henryk, On the problem of Jacobsthal. Demonstratio Math. 11 (1978), 225-232. doi:10.1515/dema-1978-0121
[MaPo90] Maier, Helmut and Pomerance, Carl, Unusually large gaps between consecutive primes. Trans. Amer. Math. Soc. 322 (1990), 201-237.
[FGKMT18] Ford, Kevin and Green, Ben and Konyagin, Sergei and Maynard, James and Tao, Terence, Long gaps between primes. J. Amer. Math. Soc. 31 (2018), 65-105. arXiv:1412.5029
[OAI26a] OpenAI, A quadratic bound for Jacobsthal's function. OpenAI Math Release preprint (2026). https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/paper.pdf

Source and licence

Imported from Formal Conjectures (Erdős problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.