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<caldwell@utm.edu> In 1905, the Reverend Cullen was interested in the numbers n^{.}2^{n}+1 (denoted C_{n}). He noticed that the first, C_{1}=3, was prime, but with the possible exception of the 53rd, the next 99 were all composite. Very soon afterwards, Cunningham discovered that 5591 divides C_{53}, and noted these numbers are composite for all n in the range 2 < n < 200, with the possible exception of 141. Five decades later Robinson showed C_{141} was a prime. These numbers are now called the Cullen numbers. Sometimes, the name "Cullen number" is extended to also include the Woodall numbers: W_{n}=n^{.}2^{n}1 (then these are then the "Cullen primes of the second kind"). A Cullen prime is any prime of the form C_{n}. The only known Cullen primes C_{n} are those with n=1, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, 361275, 481899, 1354828, 6328548, and 6679881. It has been shown that almost all Cullen numbers C_{n} are composite! Fermat's little theorem tells us if p is an odd prime, then p divides both C_{p1}, C_{p2} (and more generally, C_{m(k)} for each m(k) = (2^{k}k)(p1)k, k > 0). It has also been shown that the prime p divides C_{(p+1)/2} whenever the Jacobi symbol (2p) is 1, and p divides C_{(3p1)/2} whenever the Jacobi symbol (2p) is +1. Still it has been conjectured that there are infinitely many Cullen primes C_{n}, and it is not yet known if n and C_{n} can be simultaneously prime. 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, so any prime that can be written in this form could be called a generalized Cullen prime. We emphasize can be because at first glance neither of the following have the correct form: 669^{.}2^{128454}+1, 755^{.}2^{48323}+1But these two primes may be written as follows: 42816^{.}8^{42816}+1 and 6040^{.}256^{6040}+1 (respectively).
See Also: WoodallNumber, Fermats, Mersennes Related pages (outside of this work)
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Chris K. Caldwell © 19992014 (all rights reserved)
