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The Heegner numbers are 1, 2, 3, 7, 11, 19, 43, 67, 163. The ring of integers $\textbf{Q}(\sqrt{-d})$ have unique factorizations.
1 gives the Gaussian integers.
3 gives the Eisenstein integers.
7 gives the Kleinian integers.

What happened to 2, 11, 19, and the others? Here are pictures of the primes for 1, 2, 3, 7.

Heegner primes

Ed Pegg
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  • Mathematica, and I basically used a complex number version of the Sieve of Eratosthenes. – Ed Pegg Jul 11 '16 at 23:52
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    What happened is that a lot of professional mathematicians consider literal numbers to be beneath them. At least with $\mathbb{Z}[i]$ and $\mathbb{Z}[\omega]$, this evil is mitigated by the fact that the lattice shapes are equilateral, and therefore more elegant than in the other domains. – Robert Soupe Jul 12 '16 at 01:55
  • Also, there are practical considerations. By the time you get to $d = 163$, each purely real integer is $\sqrt{41}$ away from the nearest complex integer in the domain. – Robert Soupe Jul 12 '16 at 01:57

1 Answers1

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I've decided to call $\mathbb{Q}(\sqrt{-2})$ the Hippasus integers, after Hippasus, who was murdered for proving $\sqrt{2}$'s irrationality.

Ed Pegg
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    I like this idea lmao. I’ll start using it when I can. Do you want to say instead the Hipassus Rationals and Hippasus integers? – Sidharth Ghoshal Oct 27 '23 at 23:16