Siegel's conjecture

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.

Define Bernoulli numbers by the formal power series $t/(e^t-1)=\sum_{m=0}^{\infty}B_mt^m/m!$. An odd prime $p$ is regular if $p$ divides none of the numerators, in lowest terms, of $B_2,B_4,\ldots,B_{p-3}$. For $p=3$ this list is empty. Include $2$ as regular by convention. Put $R(x)=\#\{p\le x:p\text{ is a regular prime}\}$ and $\pi(x)=\#\{p\le x:p\text{ is prime}\}$. Relative natural density among primes means the limit of $R(x)/\pi(x)$ as $x\to\infty$.

Statement recorded in the supplied source.

Are there infinitely many regular primes? Does their relative natural density exist and equal $e^{-1/2}$, that is, does $\lim_{x\to\infty}R(x)/\pi(x)=e^{-1/2}$? [S2]

Short English statement

Are there infinitely many regular primes, and do they constitute the predicted fraction of all primes?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400160 (AMR-093-0160). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, Siegel's conjecture. https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory.
• S3 William Hart, David Harvey and Wilson Ong, Irregular primes to two billion, arXiv:1605.02398v1 (2016) Introduction and Section 2, pp. 1--3. https://arxiv.org/pdf/1605.02398.
• S4 OEIS Foundation and contributors, The On-Line Encyclopedia of Integer Sequences, A007703: Regular primes., official data snapshot 693b5499c9d2 (2026) Name, definition and comments. https://raw.githubusercontent.com/oeis/oeisdata/693b5499c9d248f570dd2d979e8bd1d7eecf689b/seq/A007/A007703.seq.
• S5 OEIS Foundation and contributors, The On-Line Encyclopedia of Integer Sequences, A000928: Irregular primes: primes p such that at least one of the numerators of the Bernoulli numbers B_2, B_4, ..., B_{p-3} (A000367) is divisible by p., official data snapshot 693b5499c9d2 (2026) Name, definition and comments. https://raw.githubusercontent.com/oeis/oeisdata/693b5499c9d248f570dd2d979e8bd1d7eecf689b/seq/A000/A000928.seq.