Cullen number


In mathematics, a Cullen number is a member of the natural number sequence of the form . 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 for. In that sense, almost all Cullen numbers are composite. 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:
Still, it is conjectured that there are infinitely many Cullen primes.
As of March 2020, the largest known generalized Cullen prime is 2805222*252805222+1. It has 3,921,539 digits and was discovered by Tom Greer, a PrimeGrid participant.
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 for each m =
− k. It has also been shown that the prime number p divides C / 2 when the Jacobi symbol is −1, and that p divides C / 2 when the Jacobi symbol 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.
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 and p does not divide b, then bn must be 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, n × bn + 1 is prime, then b must be divisible by 3.
Least n such that n × bn + 1 is prime are
bnumbers n such that n × bn + 1 is prime OEIS sequence
11, 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,...
21, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, 361275, 481899, 1354828, 6328548, 6679881,...
32, 8, 32, 54, 114, 414, 1400, 1850, 2848, 4874, 7268, 19290, 337590, 1183414,...
41, 3, 7, 33, 67, 223, 663, 912, 1383, 3777, 3972, 10669, 48375,...
51242, 18390,...
61, 2, 91, 185, 387, 488, 747, 800, 9901, 10115, 12043, 13118, 30981, 51496,...
734, 1980, 9898,...
85, 17, 23, 1911, 20855, 35945, 42816,..., 749130,...
92, 12382, 27608, 31330, 117852,...
101, 3, 9, 21, 363, 2161, 4839, 49521, 105994, 207777,...
1110,...
121, 8, 247, 3610, 4775, 19789, 187895,...
13...
143, 5, 6, 9, 33, 45, 243, 252, 1798, 2429, 5686, 12509, 42545,...
158, 14, 44, 154, 274, 694, 17426, 59430,...
161, 3, 55, 81, 223, 1227, 3012, 3301,...
1719650, 236418,...
181, 3, 21, 23, 842, 1683, 3401, 16839, 49963, 60239, 150940, 155928,...
196460,...
203, 6207, 8076, 22356, 151456,...
212, 8, 26, 67100,...
221, 15, 189, 814, 19909, 72207,...
234330, 89350,...
242, 8, 368,...
252805222,...
26117, 3143, 3886, 7763, 64020, 88900,...
272, 56, 23454,..., 259738,...
281, 48, 468, 2655, 3741, 49930,...
29...
301, 2, 3, 7, 14, 17, 39, 79, 87, 99, 128, 169, 221, 252, 307, 3646, 6115, 19617, 49718,...