Lucky Number of Euler
A number 
such that the Prime-Generating Polynomial
is Prime for
, 1, ..., 
. Such numbers are related to the Complex QuadraticField in which the Ring of Integers is factorable. Specifically, the Lucky numbers of Euler(excluding the trivial case 
) are those numbers such that the Quadratic Field 
hasClass Number 1 (Rabinowitz 1913, Le Lionnais 1983, Conway and Guy 1996). As 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 therequired form. Therefore, the only Lucky numbers of Euler are 2, 3, 5, 11, 17, and 41 (Le Lionnais 1983, Sloane's A014556), andthere does not exist a better Euler's 
form.
References
Conway, J. H. and Guy, R. K. ``The Nine Magic Discriminants.'' In The Book of Numbers. New York: Springer-Verlag, pp. 224-226, 1996. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, pp. 88 and 144, 1983. Rabinowitz, G. ``Eindeutigkeit der Zerlegung in Primzahlfaktoren in quadratischen Zahlkörpern.'' Proc. Fifth Internat. Congress Math. (Cambridge) 1, 418-421, 1913. Sloane, N. J. A. Sequence A014556in ``The On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html. Stark, H. M. ``A Complete Determination of the Complex Quadratic Fields of Class Number One.'' Michigan Math. J. 14, 1-27, 1967.
- Spiric Section
- Spirograph
- Spirolateral
- Spline
- Splitting
- Splitting Algorithm
- Sponge
- Sporadic Group
- Sports
- Sprague-Grundy Function
- Sprague-Grundy Number
- Sprague-Grundy Value
- Spread (Link)
- Spread (Tree)
- Spun Knot
- Squarable
- Square
- Square Bracket Polynomial
- Square Cupola
- Square Curve
- Square Cutting
- Squared
- Squared Square
- Square-Free
- Squarefree