Inmathematics, aCullen number is a member of theinteger sequence${\displaystyle C_n=n\cdot 2^n+1}$ (where${\displaystyle n}$ is anatural number). Cullen numbers were first studied byJames Cullen in 1905. The numbers are special cases ofProth numbers.

In 1976Christopher Hooley showed that thenatural density of positiveintegers${\displaystyle n\le x}$ for whichC_n is aprime is of theordero(x) for${\displaystyle x\to \mathrm{\infty }}$. In that sense,almost all Cullen numbers arecomposite.^[1] Hooley's proof was reworked by Hiromi Suyama to show that it works for any sequence of numbersn·2^{n +a} +b wherea andb are integers, and in particular also forWoodall numbers. The only knownCullen primes are those forn equal to:

1, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, 361275, 481899, 1354828, 6328548, 6679881 (sequenceA005849 in theOEIS).

Still, it isconjectured that there are infinitely many Cullen primes.

A Cullen numberC_n isdivisible byp = 2n − 1 ifp is aprime number of the form 8k − 3; furthermore, it follows fromFermat's little theorem that ifp is anodd prime, thenp dividesC_m(k) for eachm(k) = (2^k − k)  (p − 1) − k (fork > 0). It has also been shown that the prime numberp dividesC_{(p + 1)/2} when theJacobi symbol (2 | p) is −1, and thatp dividesC_{(3p − 1)/2} when the Jacobi symbol (2 | p) is + 1.

It is unknown whether there exists a prime numberp such thatC_p is also prime.

C_p follows therecurrence relation

${\displaystyle C_p=4(C_{p-1}+C_{p-2})+1}$.

Sometimes, ageneralized Cullen number baseb is defined to be a number of the formn·bⁿ + 1, wheren + 2 > b; if a prime can be written in this form, it is then called ageneralized Cullen prime.Woodall numbers are sometimes calledCullen numbers of the second kind.^[2]

As of October 2021, the largest known generalized Cullen prime is 2525532·73^2525532 + 1. It has 4,705,888 digits and was discovered by Tom Greer, aPrimeGrid participant.^[3][4]

According toFermat's little theorem, if there is a primep such thatn is divisible byp − 1 andn + 1 is divisible byp (especially, whenn =p − 1) andp does not divideb, thenbⁿ must becongruent to 1 modp (sincebⁿ is a power ofb^{p − 1} andb^{p − 1} is congruent to 1 modp). Thus,n·bⁿ + 1 is divisible byp, so it is not prime. For example, if somen congruent to 2 mod 6 (i.e. 2, 8, 14, 20, 26, 32, ...),n·bⁿ + 1 is prime, thenb must be divisible by 3 (exceptb = 1).

The leastn such thatn·bⁿ + 1 is prime (with question marks if this term is currently unknown) are^[5][6]

1, 1, 2, 1, 1242, 1, 34, 5, 2, 1, 10, 1, ?, 3, 8, 1, 19650, 1, 6460, 3, 2, 1, 4330, 2, 2805222, 117, 2, 1, ?, 1, 82960, 5, 2, 25, 304, 1, 36, 3, 368, 1, 1806676, 1, 390, 53, 2, 1, ?, 3, ?, 9665, 62, 1, 1341174, 3, ?, 1072, 234, 1, 220, 1, 142, 1295, 8, 3, 16990, 1, 474, 129897, ?, 1, 13948, 1, ?, 3, 2, 1161, 12198, 1, 682156, 5, 350, 1, 1242, 26, 186, 3, 2, 1, 298, 14, 101670, 9, 2, 775, 202, 1, 1374, 63, 2, 1, ... (sequenceA240234 in theOEIS)

• Cullen, James (December 1905), "Question 15897",Educ. Times: 534.
• Guy, Richard K. (2004),Unsolved Problems in Number Theory (3rd ed.), New York:Springer Verlag, Section B20,ISBN 0-387-20860-7,Zbl 1058.11001.
• Hooley, Christopher (1976),Applications of sieve methods, Cambridge Tracts in Mathematics, vol. 70,Cambridge University Press, pp. 115–119,ISBN 0-521-20915-3,Zbl 0327.10044.
• Keller, Wilfrid (1995),"New Cullen Primes"(PDF),Mathematics of Computation,64 (212):1733–1741,S39 –S46,doi:10.2307/2153382,ISSN 0025-5718,JSTOR 2153382,Zbl 0851.11003.

• Chris Caldwell,The Top Twenty: Cullen primes at ThePrime Pages.
• The Prime Glossary: Cullen number at The Prime Pages.
• Chris Caldwell,The Top Twenty: Generalized Cullen at The Prime Pages.
• Weisstein, Eric W."Cullen number".MathWorld.
• Cullen prime: definition and status (outdated), Cullen Prime Search is now hosted atPrimeGrid
• Paul Leyland,(Generalized) Cullen and Woodall Numbers

• Integer sequences
• Unsolved problems in number theory
• Classes of prime numbers

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