Rudin's conjecture on squares in progressions

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For positive integers $N,q,a$, define \[ Q(N;q,a)=\#\{j\in\{0,\ldots,N-1\}: a+jq=t^2\text{ for some integer }t\}, \qquad Q(N)=\max_{q,a\ge1}Q(N;q,a). \] This maximum exists because the possible counts belong to the finite set $\{0,\ldots,N\}$. The progression with $q=24$ and $a=1$ gives the comparison value $Q(N;24,1)$.

Statement recorded in the supplied source.

Prove the bound $Q(N)=O(\sqrt N)$: there should exist an absolute constant $C>0$ such that $Q(N)\le C\sqrt N$ for all $N\ge1$. In the stronger form, prove \[ Q(N)=Q(N;24,1)\qquad\text{for every integer }N>6. \] Both parts concern arithmetic progressions with positive starting term and positive common difference, as in this catalogue record. [S2]

Short English statement

Bound the number of squares in any $N$-term positive arithmetic progression by a constant times $\sqrt N$. The stronger conjecture says that, for $N>6$, the progression starting at $1$ with difference $24$ attains the maximum.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3200008 (AMR-031-0008). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Enrique González-Jiménez and Xavier Xarles, On a conjecture of Rudin on squares in arithmetic progressions (2013), Introduction, pp.1–3. https://arxiv.org/abs/1301.5122.