Source labels openChecked July 26, 2026

Büchi's problem

*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 $a = 0$ .

Number theory AMS 11

Mathematical statement

*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.

Statement source: Wikipedia statement material

Statement terms: CC-BY-SA-4.0

Attributed source material. Reuse must follow the linked attribution and share-alike terms.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

buchi_problem_M5

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem buchi_problem_M5 : IsBuchi 5 := by  sorry

Statement source: Formal Conjectures
Lean version: v4.27.0
Placeholder: Present; no proof artifact
Source evidence: Pinned source index
Fidelity review: Community formulation

References

Pinned Formal Conjectures sourceformalization Wikipediabackground