For a given ,
determine a complete list of fundamental binary
quadratic form discriminants such that the class number
is given by .
Heegner (1952) gave a solution for , but it was not completely accepted due to a number of apparent
gaps. However, subsequent examination of Heegner's proof showed it to be "essentially"
correct (Conway and Guy 1996). Conway and Guy (1996) therefore call the nine values
of
having
where
is the binary quadratic form discriminant
corresponding to an quadratic field (, , , , , , , , and ; OEIS A003173) the
Heegner numbers. The Heegner
numbers have a number of fascinating properties.

Stark (1967) and Baker (1966) gave independent proofs of the fact that only nine such numbers exist; both proofs were accepted. Baker (1971) and Stark (1975) subsequently
and independently solved the generalized class number problem completely for .
Oesterlé (1985) solved the case , and Arno (1992) solved the case . Wagner (1996) solved the cases , 6, and 7. Arno et al. (1993) solved the problem
for odd satisfying . Using extensive computations, Watkins (2004)
has solved the problem for all .

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Number 2." Ann. Math.94, 139-152, 1971.Conway, J. H.
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Analysis und Modulfunktionen." Math. Z.56, 227-253, 1952.Heilbronn,
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One." Quart. J. Math. (Oxford)5, 293-301, 1934.Ireland,
K. and Rosen, M. A
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p. 192, 1990.Lehmer, D. H. "On Imaginary Quadratic Fields
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