Rudin's conjecture on squares in arithmetic progressions

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

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-rudins-conjecture,
  title        = {Rudin's conjecture on squares in arithmetic progressions},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/rudins-conjecture}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

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

rudins_conjecture_strong. A stronger form of Rudin's conjecture: for every $N \geq 6$ , the arithmetic progression $24 n + 1$ attains the maximum $Q (N)$ .

rudins_conjecture_unique. The strongest form of Rudin's conjecture also asserts uniqueness: for $N \geq 6$ , any non-trivial arithmetic progression attaining the maximum $Q (N)$ has common difference $24$ . (Its initial term is then forced by $g cd (24, a) = 1$ ; the progression $24 n + 1$ is the canonical representative.)

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.RudinsConjecture (3 statements).

theorem rudins_conjecture :
    (fun N : ℕ => (Qmax N : ℝ)) =O[atTop] fun N : ℕ => Real.sqrt N

theorem rudins_conjecture_strong (N : ℕ) (hN : 6 ≤ N) : Q N 24 1 = Qmax N

theorem rudins_conjecture_unique (N : ℕ) (hN : 6 ≤ N) (q a : ℕ)
    (hqa : IsNontrivial q a) (hmax : Q N q a = Qmax N) : q = 24

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
[Ru60] Rudin, W., Trigonometric series with gaps, J. Math. Mech. 9 (1960), 203–227.
González-Jiménez, E. and Xarles, X., *On a conjecture of Rudin on squares in arithmetic progressions*, LMS J. Comput. Math. 17 (2014), 58–76.

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.