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.

**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

1999-05-25