The values of for which Quadratic Fields are uniquely factorable into factors of the form . Here, and are half-integers, except for and 2, in which case they are Integers. 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 known that 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 proof that only nine such numbers exist, but his proof was not accepted as complete at the time. Subsequent examination 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 Algebraic Integers. They also explain why Euler's Prime-Generating Polynomial is so surprisingly good at producing Primes.
See also Class Number, Discriminant (Binary Quadratic Form), Gauss's Class Number Problem, j-Function, Prime-Generating Polynomial, Quadratic Field
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. Sequence
A003173/M0827
in ``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.
© 1996-9 Eric W. Weisstein