## Cullen number< A number of the form ( n × 2^{n}) + 1, denoted
C_{n}, and named after the Reverend James Cullen
(1867–1933), an Irish Jesuit priest and schoolmaster. Cullen noticed
that the first, C_{1} = 3, was a prime
number, but with the possible exception of the 53rd, the next 99 were
all composite. Soon afterward, Cunningham discovered that 5591 divides C_{53},
and noted that all the Cullen numbers are composite
numbers for n in the range 2 < n < 200, with the possible
exception of 141. Five decades later Robinson showed that C_{141}
is a prime. Currently, the only known Cullen primes are those with n
= 1, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, 361275,
and 481899. Although the vast majority of Cullen numbers are composite,
it has been conjectured that there are infinitely many Cullen primes. Whether
n and C_{n} can simultaneously be prime
isn't known. Sometimes, the name "Cullen number" is extended to include
the Woodall numbers, W_{n} =
(n × 2^{n}) - 1. Finally, a few authors have
defined a number of the form (n × b^{n})
+ 1, with n + 2 > b, to be a generalized Cullen
number. ## Related category• TYPES OF NUMBERS | |||||

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