Cullen number

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In mathematics, a Cullen number is a natural number of the form n \cdot 2^n + 1 (written C_n). Cullen numbers were first studied by Fr. James Cullen in 1905. Cullen numbers are special cases of Proth numbers.

Properties[edit]

In 1976 Christopher Hooley showed that the natural density of positive integers n \leq x for which Cn is a prime is of the order o(x) for x\to\infty. In that sense, almost all Cullen numbers are composite.[1] Hooley's proof was reworked by Hiromi Suyama to show that it works for any sequence of numbers n · 2n+a + b where a and b are integers, and in particular also for Woodall numbers. The only known Cullen primes are those for n equal:

1, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, 361275, 481899, 1354828, 6328548, 6679881 (sequence A005849 in OEIS).

Still, it is conjectured that there are infinitely many Cullen primes.

As of August 2009, the largest known Cullen prime is 6679881 × 26679881 + 1. It is a megaprime with 2,010,852 digits and was discovered by a PrimeGrid participant from Japan.[2]

A Cullen number Cn is divisible by p = 2n − 1 if p is a prime number of the form 8k - 3; furthermore, it follows from Fermat's little theorem that if p is an odd prime, then p divides Cm(k) for each m(k) = (2k − k)   (p − 1) − k (for k > 0). It has also been shown that the prime number p divides C(p + 1) / 2 when the Jacobi symbol (2 | p) is −1, and that p divides C(3p − 1) / 2 when the Jacobi symbol (2 | p) is +1.

It is unknown whether there exists a prime number p such that Cp is also prime.

Generalizations[edit]

Sometimes, a generalized Cullen number is defined to be a number of the form n · bn + 1, where n + 2 > b; if a prime can be written in this form, it is then called a generalized Cullen prime. Woodall numbers are sometimes called Cullen numbers of the second kind.

For b = 1 to 12, the least n such that n · bn + 1 is prime are

1, 1, 2, 1, 1242, 1, 34, 5, 2, 1, 10, 1, ... (sequence A240234 in OEIS)

As of February 2012, the largest known generalized Cullen prime is 427194 × 113 427194 + 1. It has 877,069 digits and was discovered by a PrimeGrid participant from United States.[3]

References[edit]

  1. ^ Everest, Graham; van der Poorten, Alf; Shparlinski, Igor; Ward, Thomas (2003). Recurrence sequences. Mathematical Surveys and Monographs 104. Providence, RI: American Mathematical Society. p. 94. ISBN 0-8218-3387-1. Zbl 1033.11006. 
  2. ^ "The Prime Database: 6679881*2^6679881+1", Chris Caldwell's The Largest Known Primes Database, retrieved December 22, 2009 
  3. ^ "The Prime Database: 427194 · 113^427194 + 1", Chris Caldwell's The Largest Known Primes Database, retrieved January 30, 2012 

Further reading[edit]

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