The values of for which imaginary quadratic fields are uniquely factorable into factors of the form . 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 binary quadratic form discriminants which have class number 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 . 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, , and the algebraic integers. They also explain why Euler's prime-generating polynomial is so surprisingly good at producing primes.

 

Binary Quadratic Form Discriminant, Class Number, Gauss's Class Number Problem, j-Function, Prime-Generating Polynomial, Quadratic Field, Ramanujan Constant