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.

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-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 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem buchi_problem :
    answer(sorry) ↔ ∃ 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 e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.