is prime for , 2, ..., . Such numbers are related to the imaginary
quadratic field in which the ring of integers
is factorable. Specifically, the lucky numbers of Euler (excluding the trivial case
)
are those numbers
such that the imaginary quadratic field has class
number 1 (Rabinowitz 1913, Le Lionnais 1983, Conway and Guy 1996, Ribenboim 2000).
As proved by Heegner (1952)--although his proof was not accepted as complete at the time--and subsequently established by Stark (1967), there are only nine numbers such that (the Heegner numbers, , , , , , , and ), and of these, only 7, 11, 19, 43, 67, and 163 are of
the required form. Therefore, the only lucky numbers of Euler are 2, 3, 5, 11, 17,
and 41 (Le Lionnais 1983, OEIS A014556), and
there does not exist a better prime-generating
polynomial of Euler's form.
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.Heegner,
K. "Diophantische Analysis und Modulfunktionen." Math. Z.56,
227-253, 1952.Le Lionnais, F. Les
nombres remarquables. Paris: 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
On-Line Encyclopedia of Integer Sequences."Stark, H. M. "A
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 the imaginary
quadratic field in which the ring of integers
is factorable. Specifically, the lucky numbers of Euler (excluding the trivial case
)
are those numbers 
such that the imaginary quadratic field 
has class
number 1 (Rabinowitz 1913, Le Lionnais 1983, Conway and Guy 1996, Ribenboim 2000).
such that 
(the Heegner numbers 
, 
, 
, 
, 
, 
, 
, and 
), and of these, only 7, 11, 19, 43, 67, and 163 are of
the required form. Therefore, the only lucky numbers of Euler are 2, 3, 5, 11, 17,
and 41 (Le Lionnais 1983, OEIS A014556), and
there does not exist a better prime-generating
polynomial of Euler's form.
