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I read here

The primes $p$ of the form $p = -(4a^3 + 27b^2)$

that

" It is known that the number of imaginary quadratic fields of class number 3 is finite. "

But the links did not show it.

And I know many class number questions are open for quadratic fields.

So is that claim correct ?

Reference or proof ?

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    $\begingroup$ See, e.g., ams.org/journals/mcom/2004-73-246/S0025-5718-03-01517-5/… $\endgroup$ Commented Aug 13, 2023 at 20:24
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    $\begingroup$ More generally, Gauss's conjecture that the class number of $\mathbb{Q}(\sqrt{d})$ goes to $\infty$ as $d\to-\infty$ was proven by Heilbronn in 1934, and implies that there are only finitely many imaginary quadratic number fields with any given class number. See Wikipedia. $\endgroup$ Commented Aug 13, 2023 at 20:32
  • $\begingroup$ @ArturoMagidin thanks. $\endgroup$ Commented Aug 13, 2023 at 20:39
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    $\begingroup$ zakuski.math.utsa.edu/~jagy/Hudson_Williams_1991.pdf $\endgroup$ Commented Aug 13, 2023 at 21:43
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    $\begingroup$ Possibly more shelf-stable information for the future: the papers cited in the comments above are M. Watkins, "Class numbers of imaginary quadratic fields," in Math. Comp. 73 (2004), 907-938, doi.org/10.1090/S0025-5718-03-01517-5 and K. Williams & R. Hudson, "Representation of primes by the principal form of discriminant -D when the classnumber h(-D) is 3," in Acta Arithmetica 57 (1991), 131-153, doi.org/10.4064/aa-57-2-131-153 (and the second link even leads to a nicer-looking PDF) :) $\endgroup$ Commented Aug 13, 2023 at 22:05

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