Imaginary quadratic fields with a prescribed class number

Gauss's class number problem asks which imaginary quadratic fields have a given class number. It is conjectured that every positive integer $h$ is the class number of some such field, yet nobody knows how to find one quickly when $h$ is large. The task is an algorithm that does so for $h$ between $2^{90}$ and $2^{100}$.

Problem

For a negative fundamental discriminant $D$, let $h(D)$ be the class number of $\mathbb{Q}(\sqrt{D})$.

Task. Give an algorithm that, on input a positive integer $h$, returns a negative fundamental discriminant $D$ with $h(D) = h$ and $|D| \le 10^5 h^2$, fast enough to succeed within minutes for $h$ between $2^{90}$ and $2^{100}$.

Background

Let $F(h)$ be the number of imaginary quadratic fields with class number $h$. Watkins determined $F(h)$ unconditionally for all $h \le 100$ [3], completing a line of work on Gauss's problem. Soundararajan established the average size of $F(h)$ and speculated about its growth [1], and Holmin, Jones, Kurlberg, McLeman and Petersen refined this into a conjectural asymptotic formula, also studying which abelian groups occur as class groups [2]. In particular, these authors conjecture that every $h$ occurs. Since $h(D)$ grows roughly like $\sqrt{|D|}$, a field with a huge prescribed class number has a huge discriminant, and Epoch designed the challenge so that passing it should require a genuinely good idea.

What is known

• $F(h)$ is known exactly for $h \le 100$ (Watkins) [3].
• The average of $F(h)$ is known, and its typical size is conjectured [1, 2]; that $F(h) > 0$ for every $h$ is not proved.
• Epoch's verifier assumes the generalized Riemann hypothesis when it certifies class numbers at this size.

What would count as progress

• A program that, given $h$, outputs $D$; each answer is checked by computing $h(D)$ exactly.
• Epoch scores submissions with its own verifier: a Python or Sage program receives $h$ and prints $D$. It is run on up to 100 challenges within 30 minutes (the submitter may fix the range $2^s \le h < 2^{s+1}$, scored only for $s \ge 90$), and a correct answer to any challenge counts.
• An algorithm proved always to succeed would also prove that every positive integer is the class number of an imaginary quadratic field.

References

1. K. Soundararajan, The number of imaginary quadratic fields with a given class number, 2007. arXiv:0707.0237
2. S. Holmin, N. Jones, P. Kurlberg, C. McLeman and K. L. Petersen, Missing class groups and class number statistics for imaginary quadratic fields, Experiment. Math. 28 (2019), 233–254. doi:10.1080/10586458.2017.1383952
3. M. Watkins, Class numbers of imaginary quadratic fields, Math. Comp. 73 (2004), 907–938. doi:10.1090/S0025-5718-03-01517-5

Source

Imported from FrontierMath: Open Problems, quadratic-field-class-number, curated by Epoch AI. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.