As proved by Heegner (1952)--although his proof was not accepted as complete at the time--and subsequently established by Stark (1967), among positive squarefree
integers ,
the imaginary quadratic field has class number
1 for exactly the nine Heegner numbers, 2, 3, 7, 11, 19, 43, 67, and 163. Of these, precisely 7,
11, 19, 43, 67, and 163 have the form with . Therefore, the only lucky numbers of Euler are 2, 3,
5, 11, 17, and 41 (Le Lionnais 1983, OEIS A014556),
and the longest initial run of prime values among
these prime-generating polynomials
has 40 terms, attained at .
Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. Experimental
Mathematics in Action. Wellesley, MA: A K Peters, p. 13, 2007.Conway,
J. H. and Guy, R. K. "The Nine Magic Discriminants." In The
Book of Numbers. New York: Springer-Verlag, pp. 224-226, 1996.Granville,
A. and Mollin, R. A. "Rabinowitsch Revisited." Acta Arith.96,
139-153, 2000. https://doi.org/10.4064/aa96-2-3.Heegner,
K. "Diophantische Analysis und Modulfunktionen." Math. Z.56,
227-253, 1952.Le Lionnais, F. Les
nombres remarquables. Paris, France: Hermann, pp. 88 and 144, 1983.Meyer,
C. "Bemerkungen zum Satz von Heegner-Stark über die imaginär-quadratischen
Zahlkörper mit der Klassenzahl Eins." J. reine angew. Math.242,
179-214, 1970.Rabinowitz, G. "Eindeutigkeit der Zerlegung in Primzahlfaktoren
in quadratischen Zahlkörpern." Proc. Fifth Internat. Congress Math.
(Cambridge)1, 418-421, 1913.Ribenboim, P. My
Numbers, My Friends. New York: Springer-Verlag, 2000.Sloane,
N. J. A. Sequence A014556 in "The
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Complete Determination of the Complex Quadratic Fields of Class Number One."
Michigan Math. J.14, 1-27, 1967.
such that the prime-generating
polynomial
, 2, ..., 
. Such numbers are related to imaginary
quadratic fields whose rings of integers are
unique factorization domains. Specifically,
is a lucky number of Euler iff 
is squarefree and the imaginary quadratic field 
has class number
1 (Rabinowitz 1913, Le Lionnais 1983, Conway and Guy 1996, Granville and Mollin 2000,
Ribenboim 2000).
,
the imaginary quadratic field 
has class number
1 for exactly the nine Heegner numbers 
, 2, 3, 7, 11, 19, 43, 67, and 163. Of these, precisely 7,
11, 19, 43, 67, and 163 have the form 
with 
. Therefore, the only lucky numbers of Euler are 2, 3,
5, 11, 17, and 41 (Le Lionnais 1983, OEIS A014556),
and the longest initial run of prime values among
these prime-generating polynomials
has 40 terms, attained at 
.
