A Cunningham number is a binomial number of the form 
with 
and 
positive integers. 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 
.
is always divisible by 3 when 
is odd.
Primes of the form 
are also very rare. The Mersenne
numbers 
are known to be prime only for 44 values, the first few of which are 
, 3, 5, 7, 13, 17, 19, ... (OEIS A000043).
Such numbers are known as Mersenne primes. 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 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.
,
"
Report NM-R9212, Centrum voor Wiskunde en Informatica. Amsterdam, June 1992.
."
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