The values of 
for which imaginary 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 binary quadratic
form 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 
(OEIS A003173), corresponding
to discriminants 
,
, 
, 
, 
, 
, 
, 
, and 
, respectively. This was proved by Heegner (1952)--although
his proof was not accepted as complete at the time (Meyer 1970)--and subsequently
established by Stark (1967).
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.
