Editing Heegner number (section)

{{Short description|Concept in algebraic number theory}}
In [[number theory]], a '''Heegner number''' (as termed by [[John Horton Conway|Conway]] and [[Richard K. Guy|Guy]]) is a [[Square-free integer|square-free positive integer]] ''d'' such that the imaginary [[quadratic field]] <math>\Q\left[\sqrt{-d}\right]</math> has [[ideal class group|class number]] 1. Equivalently, the [[ring of integers|ring of algebraic integers]] of <math>\Q\left[\sqrt{-d}\right]</math> has [[unique factorization]].<ref>{{cite book
  | last = Conway
  | first = John Horton
  | authorlink = John Horton Conway
  | author2 = Guy, Richard K.
  | title = The Book of Numbers
  | publisher = Springer
  | year = 1996
  | page = [https://archive.org/details/bookofnumbers0000conw/page/224 224]
  | isbn = 0-387-97993-X
  | url = https://archive.org/details/bookofnumbers0000conw/page/224
  }}
</ref>

The determination of such numbers is a special case of the [[class number problem]], and they underlie several striking results in number theory.

According to the (Baker–)[[Stark–Heegner theorem]] there are precisely nine Heegner numbers:
{{block indent|left=1.6|1, 2, 3, 7, 11, 19, 43, 67, and 163. {{OEIS|A003173}}}}
This result was conjectured by [[Carl Friedrich Gauss|Gauss]] and proved up to minor flaws by [[Kurt Heegner]] in 1952. [[Alan Baker (mathematician)|Alan Baker]] and [[Harold Stark]] independently proved the result in 1966, and Stark further indicated that the gap in Heegner's proof was minor.<ref>{{citation|last=Stark|first=H. M.|authorlink=Harold Stark|year=1969|url=http://deepblue.lib.umich.edu/bitstream/2027.42/33039/1/0000425.pdf|title=On the gap in the theorem of Heegner|journal=[[Journal of Number Theory]]|volume=1|issue=1|pages=16&ndash;27|doi=10.1016/0022-314X(69)90023-7|bibcode=1969JNT.....1...16S|hdl=2027.42/33039|hdl-access=free}}</ref>