Heegner numbers

This article page is a stub, please help by expanding it.

The Heegner numbers are the nine integers which correspond to the only imaginary quadratic integer rings which are unique factorization domains, namely:

–1, –2, –3, –7, –11, –19, –43, –67, –163

(multiplied by –1, these are listed in A003173; see also A061574). These correspond to $\mathbb {Z} [i]$, $\mathbb {Z} [{\sqrt {-2}}]$, $\mathbb {Z} [\omega ]$, ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-7}})}$, ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-11}})}$, ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-19}})}$, ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-43}})}$, ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-67}})}$ and ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-163}})}$.

In 1952, Kurt Heegner proved this list is complete, but his proof was thought to be wrong. In 1968, Harold Stark demonstrated that the problem with Heegner's proof was minor and easily resolved. Specifically, Heegner had jumped to a conclusion about the reducibility of a 24th degree polynomial, a conclusion which Stark and others proved was correct.

The corresponding question for real quadratic integer rings remains so far unsolved.

See also

• Ramanujan constant

References

• Harold M. Stark, "A Complete Determination of the Complex Quadratic Fields of Class Number One", Michigan Math. J. 14, 1-27, 1967.
• Harold M. Stark, "On the 'Gap' in a Theorem of Heegner", J. of Number Theory 1 (1969): 16-27.
• Ian Stewart & David Tall, Algebraic Number Theory and Fermat's Last Theorem, 3rd Ed. Natick, Massachusetts: A. K. Peters (2002): p. 83, Theorem 4.10.