Gauss's Class Number Problem

For a given $m$, determine a complete list of fundamental Discriminants $-d$ such that the Class Number is given by $h(-d)=m$. Heegner (1952) gave a solution for $m = 1$, 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 $n(-d)$ having $h(-d)=1$ where $-d$ is the Discriminant corresponding to a Quadratic Field $a+b\sqrt{-n}$ ($n=-1$, $-2$, $-3$, $-7$, $-11$, $-19$, $-43$, $-67$, and $-163$; Sloane's 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 $m=2$. Oesterlé (1985) solved the case $m=3$, and Arno (1992) solved the case $m=4$. Wagner (1996) solve the cases $n=5$, 6, and 7. Arno et al. (1993) solved the problem for Odd $m$ satisfying $5\leq m\leq 23$.

See also Class Number, Gauss's Class Number Conjecture, Heegner Number

References

Arno, S. ``The Imaginary Quadratic Fields of Class Number 4.'' Acta Arith. 40, 321-334, 1992.

Arno, S.; Robinson, M. L.; and Wheeler, F. S. ``Imaginary Quadratic Fields with Small Odd Class Number.'' Dec. 1993. http://www.math.uiuc.edu/Algebraic-Number-Theory/0009/.

Baker, A. ``Linear Forms in the Logarithms of Algebraic Numbers. I.'' Mathematika 13, 204-216, 1966.

Baker, A. ``Imaginary Quadratic Fields with Class Number 2.'' Ann. Math. 94, 139-152, 1971.

Conway, J. H. and Guy, R. K. ``The Nine Magic Discriminants.'' In The Book of Numbers. New York: Springer-Verlag, pp. 224-226, 1996.

Goldfeld, D. M. ``Gauss' Class Number Problem for Imaginary Quadratic Fields.'' Bull. Amer. Math. Soc. 13, 23-37, 1985.

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.

Lehmer, D. H. ``On Imaginary Quadratic Fields whose Class Number is Unity.'' Bull. Amer. Math. Soc. 39, 360, 1933.

Montgomery, H. and Weinberger, P. ``Notes on Small Class Numbers.'' Acta. Arith. 24, 529-542, 1974.

Oesterlé, J. ``Nombres de classes des corps quadratiques imaginaires.'' Astérique 121-122, 309-323, 1985.

Oesterlé, J. ``Le problème de Gauss sur le nombre de classes.'' Enseign Math. 34, 43-67, 1988.

Serre, J.-P. $\Delta=b^2-4ac$.'' Math. Medley 13, 1-10, 1985.

Shanks, D. ``On Gauss's Class Number Problems.'' Math. Comput. 23, 151-163, 1969.

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.

Stark, H. M. ``A Complete Determination of the Complex Quadratic Fields of Class Number One.'' Michigan Math. J. 14, 1-27, 1967.

Stark, H. M. ``On Complex Quadratic Fields with Class Number Two.'' Math. Comput. 29, 289-302, 1975.

Wagner, C. ``Class Number 5, 6, and 7.'' Math. Comput. 65, 785-800, 1996.