Gauss's class number problem asks, for a given positive integer, for a complete list of fundamental binary
quadratic form discriminants such that the class number
is , with .
Heegner (1952) gave a solution for ,
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 positive squarefree
values , 2, 3, 7, 11, 19, 43, 67, and 163 (OEIS A003173)
for which the imaginary quadratic field
has class number 1 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 . Oesterlé (1985) solved the case , and Arno (1992) solved the case . Wagner (1996) solved the cases , 6, and 7. Arno et al. (1998) solved the problem
for odd satisfying .
Using extensive computations, Watkins (2004) has solved the problem for all .
Kimura (2026) reports a complete unconditional list for every .
Arno, S. "The Imaginary Quadratic Fields of Class Number 4." Acta Arith.60, 321-334, 1992.Arno, S.; Robinson,
M. L.; and Wheeler, F. S. "Imaginary Quadratic Fields with Small Odd
Class Number." Acta Arith.83, 295-330, 1998. https://doi.org/10.4064/aa-83-4-295-330.Baker,
A. "Linear Forms in the Logarithms of Algebraic Numbers. I." Mathematika13,
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.Ireland,
K. and Rosen, M. A
Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer-Verlag,
p. 192, 1990.Kimura, I. "Imaginary Quadratic Fields of Class
Number at Most 2000." 15 Oct 2026. https://arxiv.org/abs/2610.21511.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." Astérisque121-122, 309-323,
1985.Oesterlé, J. "Le problème de Gauss sur le nombre
de classes." Enseign Math.34, 43-67, 1988.Serre,
J.-P. "."
Math. Medley13, 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 "The On-Line Encyclopedia of Integer Sequences."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.Watkins,
M. "Class Numbers of Imaginary Quadratic Fields." Math. Comput.73,
907-938, 2004.
, for a complete list of fundamental binary
quadratic form discriminants 
such that the class number
is 
, with 
.
Heegner (1952) gave a solution for 
,
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 positive squarefree
values 
, 2, 3, 7, 11, 19, 43, 67, and 163 (OEIS A003173)
for which the imaginary quadratic field 
has class number 1 the Heegner
numbers. The Heegner numbers have a number
of fascinating properties.
. Oesterlé (1985) solved the case 
, and Arno (1992) solved the case 
. Wagner (1996) solved the cases 
, 6, and 7. Arno et al. (1998) solved the problem
for odd 
satisfying 
.
Using extensive computations, Watkins (2004) has solved the problem for all 
.
.
."
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