Cunningham Number
A Binomial Number of the form 
. Bases 
which are themselves powers need not beconsidered since they correspond to 
. Prime Numbers of the form
are very rare.
A Necessary (but not Sufficient) condition for 
to be Prime is that 
be of the form
. Numbers of the form 
are called Fermat Numbers, and the onlyknown Primes occur for 
, 
, 
, 
, and 
(i.e., 
, 1, 2,3, 4). The only other Primes 
for nontrivial 
and 
are 
, 
,and 
.
Primes of the form 
are also very rare. The Mersenne Numbers 
areknown to be prime only for 37 values, the first few of which are 
, 3, 5, 7, 13, 17, 19, ... (Sloane's A000043). There are no otherPrimes for nontrivial 
and .
In 1925, Cunningham and Woodall (1925) gathered together all that was known about the Primality andfactorization of the numbers and published a small book of tables. These tables collected from scattered sourcesthe known prime factors for the bases 2 and 10 and also presented the authors' results of 30 years' work with these and otherbases.
Since 1925, many people have worked on filling in these tables. D. H. Lehmer, a well-known mathematician who died in 1991,was for many years a leader of these efforts. Lehmer was a mathematician who was at the forefront of computing as modernelectronic computers became a reality. He was also known as the inventor of some ingenious pre-electronic computingdevices specifically designed for factoring numbers.
Updated factorizations were published in Brillhart et al. (1988). The current archive of Cunningham number factorizationsfor 
, ..., 
is kept on ftp://sable.ox.ac.uk/pub/math/cunningham. The tables have been extended by Brent and te Riele (1992) to 
, ...,100 with 
for 
and 
for 
. All numbers with exponent 58 and smaller, and all composites with
digits have now been factored.
References
Brent, R. P. and te Riele, H. J. J. ``Factorizations of 
, 
'' Report NM-R9212, Centrum voor Wiskunde en Informatica. Amsterdam, June 1992. ftp://sable.ox.ac.uk/pub/math/factors/.
Brillhart, J.; Lehmer, D. H.; Selfridge, J.; Wagstaff, S. S. Jr.; and Tuckerman, B. Factorizations of 
, 
, 
Up to High Powers, rev. ed. Providence, RI: Amer. Math. Soc., 1988. Updates are available electronically from ftp://sable.ox.ac.uk/pub/math/cunningham/.
Cunningham, A. J. C. and Woodall, H. J. Factorisation of 
, 
, 3, 5, 6, 7, 10, 11, 12 Up to High Powers (). London: Hodgson, 1925.
Mudge, M. ``Not Numerology but Numeralogy!'' Personal Computer World, 279-280, 1997.
Ribenboim, P. ``Numbers 
.'' §5.7 in The New Book of Prime Number Records. New York: Springer-Verlag, pp. 355-360, 1996.
Sloane, N. J. A. SequenceA000043/M0672in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.
- Snake Oil Method
- Snake Polyiamond
- Snedecor's F-Distribution
- Snellius-Pothenot Problem
- Snowflake
- Snub Cube
- Snub Cuboctahedron
- Snub Disphenoid
- Snub Dodecadodecahedron
- Snub Dodecahedron
- Snub Icosidodecadodecahedron
- Snub Icosidodecahedron
- Snub Polyhedron
- Snub Square Antiprism
- Soap Bubble
- Soccer Ball
- Sociable Numbers
- Social Choice Theory
- Socrates' Paradox
- Soddy Circles
- Soddy Line
- Soddy Points
- Soddy's Hexlet
- Sofa Constant
- Soldner's Constant