Rudin's conjecture on squares in arithmetic progressions

formal-conjectures · wikipedia · ams-11

**Rudin's conjecture.** The maximal number of squares among the first $N$ terms of a non-trivial
arithmetic progression grows at most like $\sqrt{N}$:
$$Q(N) = O(\sqrt{N}).$$

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/Rudin%27s_conjecture
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/Wikipedia/RudinsConjecture.lean
Local target: Corpus.WikipediaRudinsConjecture.rudins_conjecture
Source SHA-256: a98ec6b8a3d678f437effd8a1483c1ed44dd6a69a03286119ce66f2f4fecc67e
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