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Cullen number

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Cullen number

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

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

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.

According to Fermat 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[3]

1, 1, 2, 1, 1242, 1, 34, 5, 2, 1, 10, 1, ... (sequence A240234 in 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, ... 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, ...
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 September 2015, the largest known generalized Cullen prime is 427194 × 113427194 + 1. It has 877,069 digits and was discovered by a PrimeGrid participant from United States.[4]

References

  1. ^ Everest, Graham;  
  2. ^ "The Prime Database: 6679881*2^6679881+1", Chris Caldwell's The Largest Known Primes Database, retrieved December 22, 2009 
  3. ^ List of generalized Cullen primes
  4. ^ "The Prime Database: 427194 · 113^427194 + 1", Chris Caldwell's The Largest Known Primes Database, retrieved January 30, 2012 

Further reading

  • Cullen, James (December 1905), "Question 15897", Educ. Times: 534 .
  • .  
  • .  
  • Keller, Wilfrid (1995), "New Cullen Primes" (PDF), .  

External links

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  • Cullen prime: definition and status (outdated), Cullen Prime Search is now hosted at PrimeGrid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