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Cullen number


A number of the form (n × 2n) + 1, denoted Cn, and named after the Reverend James Cullen (1867–1933), an Irish Jesuit priest and schoolmaster. Cullen noticed that the first, C1 = 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 C53, 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 C141 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 Cn can simultaneously be prime isn't known. Sometimes, the name "Cullen number" is extended to include the Woodall numbers, Wn = (n × 2n) - 1. Finally, a few authors have defined a number of the form (n × bn) + 1, with n + 2 > b, to be a generalized Cullen number.

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