In mathematics, a Cullen number is a member of the integer sequence (where is a natural number). Cullen numbers were first studied by James Cullen in 1905. The numbers are special cases of Proth numbers.

Properties

In 1976 Christopher Hooley showed that the natural density of positive integers for which Cn is a prime is of the order o(x) for . 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 to:

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.

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.[2]

As of October 2021, the largest known generalized Cullen prime is 2525532 · 732525532 + 1. It has 4,705,888 digits and was discovered by Tom Greer, a PrimeGrid participant.[3][4]

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).

The least n such that n · 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, ... (sequence A240234 in the OEIS)
b numbers n such that n × bn + 1 is prime (these n are checked up to 101757) 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, 1183414, ... 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 2805222, ...
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, ...


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. ^ Marques, Diego (2014). "On Generalized Cullen and Woodall Numbers That are Also Fibonacci Numbers" (PDF). Journal of Integer Sequences. 17.
  3. ^ "PrimeGrid Official Announcement" (PDF). Primegrid. 28 August 2021. Retrieved 14 November 2021.
  4. ^ "PrimePage Primes: 2525532 · 73^2525532 + 1". primes.utm.edu. Retrieved 14 November 2021.
  5. ^ Löh, Günter (6 May 2017). "Generalized Cullen primes".
  6. ^ Harvey, Steven (6 May 2017). "List of generalized Cullen primes base 101 to 10000".

Further reading