A Binomial Number of the form . Bases which are themselves powers need not be considered 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 only known 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 are known 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 other Primes for nontrivial and .

In 1925, Cunningham and Woodall (1925) gathered together all that was known about the Primality and factorization of the numbers and published a small book of tables. These tables collected from scattered sources the known prime factors for the bases 2 and 10 and also presented the authors' results of 30 years' work with these and other bases.

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 modern electronic computers became a reality. He was also known as the inventor of some ingenious pre-electronic computing devices specifically designed for factoring numbers.

Updated factorizations were published in Brillhart *et al. *(1988). The current archive of Cunningham number factorizations
for , ..., 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. Sequence
A000043/M0672
in ``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.

© 1996-9

1999-05-25