Imaginary quadratic fields with a prescribed class number

Give an algorithm that, for h between 2^90 and 2^100, quickly finds an imaginary quadratic field with class number exactly h.

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1	# Imaginary quadratic fields with a prescribed class number
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3	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 $2^{100}$.
4
5	## Problem
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7	For a negative fundamental discriminant $D$, let $h(D)$ be the class number of $\mathbb{Q}(\sqrt{D})$.
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9	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 $2^{100}$.
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11	## Background
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13	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.
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15	## What is known
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17	- $F(h)$ is known exactly for $h \le 100$ (Watkins) [3].
18	- 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.
19	- Epoch's verifier assumes the generalized Riemann hypothesis when it certifies class numbers at this size.
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21	## What would count as progress
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23	- A program that, given $h$, outputs $D$; each answer is checked by computing $h(D)$ exactly.
24	- 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.
25	- An algorithm proved always to succeed would also prove that every positive integer is the class number of an imaginary quadratic field.
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27	## References
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29	1. K. Soundararajan, The number of imaginary quadratic fields with a given class number, 2007. [arXiv:0707.0237](https://arxiv.org/abs/0707.0237)
30	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](https://doi.org/10.1080/10586458.2017.1383952)
31	3. M. Watkins, Class numbers of imaginary quadratic fields, Math. Comp. 73 (2004), 907–938. [doi:10.1090/S0025-5718-03-01517-5](https://doi.org/10.1090/S0025-5718-03-01517-5)
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33	## Source
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35	Imported from [FrontierMath: Open Problems, quadratic-field-class-number](https://epoch.ai/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.