Heegner Number
The values of 
for which Quadratic Fields 
are uniquely factorable intofactors of the form 
. Here, 
and 
are half-integers, except for 
and 2, in which case they areIntegers. The Heegner numbers therefore correspond to Discriminants which have Class Number 
equal to 1, except for Heegner numbers 
and 
, which correspond to 
and
, respectively.
The determination of these numbers is called Gauss's Class Number Problem, and it is now knownthat there are only nine Heegner numbers: , , 
, 
, 
, 
, 
, 
, and 
(Sloane's A003173),corresponding to discriminants 
, , , , , , , , and , respectively.
Heilbronn and Linfoot (1934) showed that if a larger 
existed, it must be 
. Heegner (1952) published a proofthat only nine such numbers exist, but his proof was not accepted as complete at the time. Subsequentexamination of Heegner's proof show it to be ``essentially'' correct (Conway and Guy 1996).
The Heegner numbers have a number of fascinating connections with amazing results in Prime Number theory.In particular, the j-Function provides stunning connections between 
, 
, and the AlgebraicIntegers. They also explain why Euler's Prime-Generating Polynomial 
is sosurprisingly good at producing Primes.
References
Conway, J. H. and Guy, R. K. ``The Nine Magic Discriminants.'' In The Book of Numbers. New York: Springer-Verlag, pp. 224-226, 1996. Heegner, K. ``Diophantische Analysis und Modulfunktionen.'' Math. Z. 56, 227-253, 1952. Heilbronn, H. A. and Linfoot, E. H. ``On the Imaginary Quadratic Corpora of Class-Number One.'' Quart. J. Math. (Oxford) 5, 293-301, 1934. Sloane, N. J. A. SequenceA003173/M0827in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.
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