Editing Heegner number (section)

==Euler's prime-generating polynomial==
Euler's [[Formula for primes#Prime formulas and polynomial functions|prime-generating polynomial]]
<math display=block>n^2 + n + 41,</math>
which gives (distinct) primes for ''n''&nbsp;=&nbsp;0,&nbsp;...,&nbsp;39, is related to the Heegner number 163&nbsp;=&nbsp;4&nbsp;·&nbsp;41&nbsp;−&nbsp;1.

[[George Yuri Rainich|Rabinowitsch]]<ref>[[George Yuri Rainich|Rabinovitch, Georg]] [https://babel.hathitrust.org/cgi/pt?id=miun.aag4063.0001.001;view=1up;seq=420 "Eindeutigkeit der Zerlegung in Primzahlfaktoren in quadratischen Zahlkörpern."] Proc. Fifth Internat. Congress Math. ( Cambridge) 1, 418–421, 1913.</ref> proved that
<math display=block>n^2 + n + p</math>
gives primes for <math>n=0,\dots,p-2</math> if and only if this quadratic's [[discriminant]] <math>1-4p</math> is the negative of a Heegner number.

(Note that <math>p-1</math> yields <math>p^2</math>, so <math>p-2</math> is maximal.)

1, 2, and 3 are not of the required form, so the Heegner numbers that work are 7, 11, 19, 43, 67, 163, yielding prime generating functions of Euler's form for 2, 3, 5, 11, 17, 41; these latter numbers are called ''[[lucky numbers of Euler]]'' by [[François Le Lionnais|F. Le Lionnais]].<ref>Le Lionnais, F. Les nombres remarquables. Paris: Hermann, pp. 88 and 144, 1983.</ref>