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Cullen Number


A Cullen number is a number of the form

 C_n=2^nn+1.

The first few are 3, 9, 25, 65, 161, 385, ... (OEIS A002064).

Cullen numbers are divisible by p=2n-1 if p is a prime of the form 8k+/-3.

The first few prime Cullen numbers are numbers 3, 393050634124102232869567034555427371542904833, ... (OEIS A050920), corresponding to n=1, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, 361275, 481899, 1354828, 6328548, 6679881, ... (OEIS A005849; Caldwell). As of Nov. 2015, it is known that there are no other Cullen primes for n<13705481 (PrimeGrid).


See also

Cunningham Number, Fermat Number, Integer Sequence Primes, Proth Number, Sierpiński Number of the First Kind, Woodall Number, Woodall Prime

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References

Caldwell, C. K. "The Top Twenty: Cullen Primes." http://primes.utm.edu/top20/page.php?id=6#records.Guy, R. K. "Cullen Numbers." §B20 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 77, 1994.Keller, W. "New Cullen Primes." Math. Comput. 64, 1733-1741, 1995.Leyland, P. http://research.microsoft.com/~pleyland/factorization/cullen_woodall/cw.htm.PrimeGrid. "PrimeGrid's Cullen Prime Search." Jul. 25, 2009. http://www.primegrid.com/download/Cullen6679881.pdf.PrimeGrid. "Subprojects: Cullen Prime Search." http://www.primegrid.com/server_status_subprojects.php.PrimeGrid. "PrimeGrid Primes: Subproject: (CUL) Cullen Prime Search." http://www.primegrid.com/primes/primes.php?project=CUL.Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 360-361, 1996.Rodenkirch, M. and Ballinger, R. "Cullen Primes: Definition and Status." http://www.prothsearch.net/cullen.html.Sloane, N. J. A. Sequences A002064/M2795, A005849/M5401, and A050920 in "The On-Line Encyclopedia of Integer Sequences."

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Cullen Number

Cite this as:

Weisstein, Eric W. "Cullen Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CullenNumber.html

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