Quadratic bound in Linnik's least-prime problem

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 integers $d\ge2$ and $1\le a<d$ with $\gcd(a,d)=1$, define \[p(a,d)=\min\{p:p\text{ is a positive prime and }p\equiv a\pmod d\}.\] Dirichlet's theorem guarantees that this set is nonempty. The minimum is taken over the entire residue class; in particular $p=a$ is allowed when $a$ is prime.

Statement recorded in the supplied source.

For every integer $d\ge2$ and every $a$ with $1\le a<d$ and $\gcd(a,d)=1$, is it true that \[p(a,d)<d^2?\] [S2]

Short English statement

Does every reduced residue class modulo d contain a prime strictly below the square of d?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400132 (AMR-093-0132). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Wikipedia contributors, Linnik's theorem, revision 1378957862 (2026) Properties: conjectural quadratic bound. https://en.wikipedia.org/w/index.php?title=Linnik%27s+theorem&oldid=1378957862.
• S3 Y. Lamzouri, X. Li and K. Soundararajan, Conditional bounds for the least quadratic non-residue and related problems, arXiv:1309.3595v3 (2016) Section 1.2, pp. 2--3: least prime in a reduced residue class. https://arxiv.org/pdf/1309.3595.
• S4 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, Quadratic bound in Linnik's least-prime problem. https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory.