Heegner Number
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.
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5th hexagonal number