Heegner Number

A Heegner number is a positive integer that is squarefree and for which the imaginary quadratic field has class number one. Equivalently, the ring of integers is a principal ideal domain and hence a unique factorization domain. For and 2, the ring of integers is

(1)

For the other Heegner numbers, all of which are congruent to 3 modulo 4, the ring of integers is

(2)

The determination of the Heegner numbers is called Gauss's class number problem. There are exactly nine: 1, 2, 3, 7, 11, 19, 43, 67, and 163 (OEIS A003173). Their corresponding fundamental discriminants are , , , , , , , , and , respectively. Heilbronn and Linfoot (1934) showed that any additional value would have to exceed . Heegner (1952) proved that the list was complete, although his proof was not accepted as complete at the time (Meyer 1970). The result was subsequently established independently by Baker (1966) and Stark (1967), and later examination showed Heegner's proof to be essentially correct (Conway and Guy 1996).

For , 7, 11, 19, 43, 67, or 163, write , where and have the same parity. The field norm of is

(3)

If is an ordinary prime number, then is a prime element of . The converse does not hold: an inert rational prime number remains a prime element of , but has . The illustrations below therefore show the elements whose norms are ordinary prime numbers, rather than all prime elements.

The elements whose norms are ordinary prime numbers for are illustrated above (Pegg 2026). Since their imaginary parts are integer multiples of , the points lie in horizontal rows. Drawing each element as a narrow rectangle makes consecutive values that are prime in a row join into the visible "barcode." The boundary circle is the norm bound used to select the plotted elements, not an additional arithmetic structure.

More specifically, if , then

(4)

For this gives , so the 40 consecutive values that are prime for , 1, ..., 39 appear as an uninterrupted bar in the first nonreal row. Replacing by gives Euler's prime-generating polynomial . The same construction for gives .

The illustration above also shows exact parabolas from Pegg's construction that contain no elements whose norms are prime. In fact, for and 3, the parabolas satisfy

(5)

Consequently, lattice points on these parabolas for which the expression on the right is an integer greater than one have square norm, so the norm is a composite number. The Wolfram Language implementation ResourceFunction["HeegnerPrime"] provides another way to compute prime elements in these rings (Pegg 2021).

The Heegner numbers also have connections with other striking results in prime number theory. In particular, the j-function provides connections between e, pi, and algebraic integers.