Heegner Number

The values of $-d$ for which Quadratic Fields $\Bbb{Q}(\sqrt{-d}\,)$ are uniquely factorable into factors of the form $a+b\sqrt{-d}$. Here, $a$ and $b$ are half-integers, except for $d=1$ and 2, in which case they are Integers. The Heegner numbers therefore correspond to Discriminants $-d$ which have Class Number $h(-d)$ equal to 1, except for Heegner numbers $-1$ and $-2$, which correspond to $d=-4$ and $-8$, 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: $-1$, $-2$, $-3$, $-7$, $-11$, $-19$, $-43$, $-67$, and $-163$ (Sloane's A003173), corresponding to discriminants $-4$, $-8$, $-3$, $-7$, $-11$, $-19$, $-43$, $-67$, and $-163$, respectively.

Heilbronn and Linfoot (1934) showed that if a larger $d$ existed, it must be $>10^9$. 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 $e$, $\pi$, and the Algebraic Integers. They also explain why Euler's Prime-Generating Polynomial $n^2-n+41$ 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.