Büchi's problem

formal-conjectures · wikipedia · ams-11

**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.

Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).

Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.

Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://en.wikipedia.org/wiki/B%C3%BCchi%27s_problem
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/Wikipedia/Buchi.lean
Local target: Corpus.WikipediaBuchi.buchi_problem_M5
Source SHA-256: 4dcdc67eed413fc6a36124561dc490ba06b8e05fb832b1cff7fc159ead7df940
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.

Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.

Public JSON record