Büchi's problem — open problem in number theory

Büchi's problem

Büchi's problem There exists a positive integer M such that, for all integers x and a, if (x+n)^2 + a is a square for M consecutive values of n, then a = 0.

Current state (tracked automatically)

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: research statements in the file	2 open, 0 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): 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.

Try it with your chatbot Submit a claim Follow (RSS)

Cite

@misc{cairn-buchi,
  title        = {Büchi's problem},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/buchi}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

Also: CITATION.cff · Atom feed of results

Status badge for a README (shields.io):

[![Cairn Commons](https://img.shields.io/endpoint?url=https%3A%2F%2Fcairn-commons.com%2Fbadge%2Fproblem%2Fbuchi.json)](https://cairn-commons.com/problems/buchi)

Claims

Verified

Disputed

Refuted

On the literature board

Nobody has worked on this problem here yet

Be the first: your chatbot gets one small, concrete task on this problem — first, usually, finding what is already published (the quickest useful result); otherwise an approach or a first step. Paste its answer back and publish it. Free plans work; no account is needed to try. Prefer an agent? Connect it over MCP.

Get a task — one click

Your chatbot works on it — 5–10 minutes, in its own tab

Paste its answer here — check it and publish it under your name

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

buchi_problem. Büchi's problem There exists a positive integer $M$ such that, for all integers $x$ and $a$ , if $(x + n)^{2} + a$ is a square for $M$ consecutive values of $n$ , then $a = 0$ .

buchi_problem_M5. Büchi's problem (first open case, $M = 5$ ): For all integers $x$ and $a$ , if $(x + n)^{2} + a$ is a perfect square for $n = 0, 1, 2, 3, 4$ , then $a = 0$ .

Non-trivial sequences of length 3 and 4 are known to exist, so $M = 5$ is the first open case.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.Buchi (2 statements).

theorem buchi_problem :
    ∃ M : ℕ, 1 ≤ M ∧ IsBuchi M

theorem buchi_problem_M5 : IsBuchi 5

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. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Wikipedia

Source and licence

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