# Cullen number

(Redirected from Cullen prime)

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

## Properties

In 1976 Christopher Hooley showed that the natural density of positive integers ${\displaystyle n\leq x}$ for which Cn is a prime is of the order o(x) for ${\displaystyle x\to \infty }$. 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 the OEIS).

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

As of February 2016, 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

Sometimes, a generalized Cullen number base b 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.[3]

According to Fermat's little theorem, if there is a prime p such that n is divisible by p - 1 and n + 1 is divisible by p (especially, when n = p - 1) and p does not divide b, then bn must be congruent to 1 mod p (since bn is a power of bp - 1 and bp - 1 is congruent to 1 mod p). Thus, n × bn + 1 is divisible by p, so it is not prime. For example, if some n congruent to 2 mod 6 (i.e. 2, 8, 14, 20, 26, 32, ...), n × bn + 1 is prime, then b must be divisible by 3 (except b = 1).

Least n such that n × bn + 1 is prime are (with question marks if this term is currently unknown)[4][5]

1, 1, 2, 1, 1242, 1, 34, 5, 2, 1, 10, 1, ?, 3, 8, 1, 19650, 1, 6460, 3, 2, 1, 4330, 2, ?, 117, 2, 1, ?, 1, 82960, 5, 2, 25, 304, 1, 36, 3, 368, 1, ?, 1, 390, 53, 2, 1, ?, 3, ?, 9665, 62, 1, ?, 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, ... (sequence A240234 in the OEIS)
 b numbers n such that n × bn + 1 is prime (these n are checked up to 100000) OEIS sequence 1 1, 2, 4, 6, 10, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70, 72, 78, 82, 88, 96, 100, 102, 106, 108, 112, 126, 130, 136, 138, 148, 150, 156, 162, 166, 172, 178, 180, 190, 192, 196, 198, 210, 222, 226, 228, 232, 238, 240, 250, 256, 262, 268, 270, 276, 280, 282, 292, ... (all primes minus 1) A006093 2 1, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, 361275, 481899, 1354828, 6328548, 6679881, ... A005849 3 2, 8, 32, 54, 114, 414, 1400, 1850, 2848, 4874, 7268, 19290, 337590, ... A006552 4 1, 3, 7, 33, 67, 223, 663, 912, 1383, 3777, 3972, 10669, 48375, ... A007646 5 1242, 18390, ... 6 1, 2, 91, 185, 387, 488, 747, 800, 9901, 10115, 12043, 13118, 30981, 51496, ... A242176 7 34, 1980, 9898, ... A242177 8 5, 17, 23, 1911, 20855, 35945, 42816, ..., 749130, ... A242178 9 2, 12382, 27608, 31330, 117852, ... A265013 10 1, 3, 9, 21, 363, 2161, 4839, 49521, 105994, 207777, ... A007647 11 10, ... 12 1, 8, 247, 3610, 4775, 19789, 187895, ... A242196 13 ... 14 3, 5, 6, 9, 33, 45, 243, 252, 1798, 2429, 5686, 12509, 42545, ... A242197 15 8, 14, 44, 154, 274, 694, 17426, 59430, ... A242198 16 1, 3, 55, 81, 223, 1227, 3012, 3301, ... A242199 17 19650, 236418, ... 18 1, 3, 21, 23, 842, 1683, 3401, 16839, 49963, 60239, 150940, 155928, ... A007648 19 6460, ... 20 3, 6207, 8076, 22356, 151456, ... 21 2, 8, 26, 67100, ... 22 1, 15, 189, 814, 19909, 72207, ... 23 4330, 89350, ... 24 2, 8, 368, ... 25 ... 26 117, 3143, 3886, 7763, 64020, 88900, ... 27 2, 56, 23454, ..., 259738, ... 28 1, 48, 468, 2655, 3741, 49930, ... 29 ... 30 1, 2, 3, 7, 14, 17, 39, 79, 87, 99, 128, 169, 221, 252, 307, 3646, 6115, 19617, 49718, ...

As of October 2016, the largest known generalized Cullen prime is 682156 × 79682156 + 1. It has 1,294,484 digits and was discovered by a PrimeGrid participant from Austria.[6]

## References

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. ^ Marques, Diego (2014). "On Generalized Cullen and Woodall Numbers That are Also Fibonacci Numbers" (PDF). Journal of Integer Sequences. 17.
4. ^ Löh, Günter (6 May 2017). "Generalized Cullen primes".
5. ^ Harvey, Steven (6 May 2017). "List of generalized Cullen primes base 101 to 10000".
6. ^ Caldwell, Chris K. "The Prime Database: 682156*79^682156+1".