Cullen's number

A Cullen number is a number of the form . The Reverend James Cullen studied these numbers in 1905 . He noticed that except for all numbers of this form to composite numbers, that is, they are not prime numbers . His uncertainty was by Allan JC Cunningham in 1906 are eliminated by this place the divider 5,591th Cunningham showed that all are composed up to n = 200, with one possible exception for n = 141. ${\displaystyle C_{n}=n\cdot 2^{n}+1}$ ${\displaystyle C_{1}=3}$ ${\displaystyle C_{99}}$ ${\displaystyle C_{53}}$ ${\displaystyle C_{n}}$

In 1958 , Raphael M. Robinson confirmed that is a prime and demonstrated that except for and all Cullen numbers from to are composite numbers. ${\displaystyle C_{141}}$ ${\displaystyle C_{1}}$ ${\displaystyle C_{141}}$ ${\displaystyle C_{1}}$ ${\displaystyle C_{1000}}$

Wilfrid Keller has 1,984 shown that and also prime numbers are, but everyone else with are numbers Cullen composite. ${\displaystyle C_{4713},C_{5795},C_{6611}}$ ${\displaystyle C_{18496}}$ ${\displaystyle C_{n}}$ ${\displaystyle n\leq 30000}$

At the moment (as of November 2015) Cullen prime numbers are known for the following : ${\displaystyle C_{n}}$ ${\displaystyle n}$

1, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, 361275, 481899, 1354828, 6328548, 6679881, ... (sequence A005849 in OEIS )

The largest known Cullen prime to date is thus , it has 2010 852 digits. It was discovered on July 25, 2009 by an anonymous Japanese participant in the PrimeGrid Internet project . ${\displaystyle C_{6679881}=6679881\cdot 2^{6679881}+1}$

It is known that there are no more prime Cullen numbers up to . It is assumed , however , that there are an infinite number of Cullen prime numbers. It is not yet known whether and may be prime at the same time. ${\displaystyle n<13705481}$ ${\displaystyle n}$ ${\displaystyle C_{n}}$

Properties of Cullen Numbers

Almost all Cullen numbers are composite numbers. They are divisible by prime numbers of the form , which must be a prime number of the form . Because of Fermat's little theorem , one can also deduce that if is an odd prime, then must be a factor of with for . ${\displaystyle p=2n-1}$ ${\displaystyle p}$ ${\displaystyle p=8k\pm 3}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle C_{m(k)}}$ ${\displaystyle m(k)=(2^{k}-k)\cdot (p-1)-k}$ ${\displaystyle k\geq 0}$

Furthermore the following could be shown:

The prime number divides the Cullen number when is Jacobi symbol . ${\displaystyle p}$ ${\displaystyle C_{\frac {p+1}{2}}}$ ${\displaystyle \left({\frac {2}{p}}\right)=-1}$

The prime number divides the Cullen number when is Jacobi symbol . ${\displaystyle p}$ ${\displaystyle C_{\frac {3p-1}{2}}}$ ${\displaystyle \left({\frac {2}{p}}\right)=+1}$

Generalized Cullen Numbers

Numbers of the form with are called generalized Cullen numbers . If such a number is a prime number, it is called a generalized Cullen prime number . ${\displaystyle n\cdot b^{n}+1}$ ${\displaystyle n+2>b}$

The smallest , for which is prime, are for ascending = 1, 2, ...: ${\displaystyle n}$ ${\displaystyle n\cdot b^{n}+1}$ ${\displaystyle b}$

1, 1, 2, 1, 1242, 1, 34, 5, 2, 1, 10, 1, ... (sequence A240234 in OEIS )

The following is a listing of the first generalized Cullen primes for bases between 1 and 30. These were examined up to at least 100,000. If the condition does not apply, but the number is still prime, it is put in brackets: ${\displaystyle b}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n+2>b}$ ${\displaystyle n\cdot b^{n}+1}$

b	n such that n • b n +1 is prime	examined up	OEIS episode
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 prime numbers minus 1)	all prime numbers	Follow A006093 in OEIS
2	1, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, 361275, 481899, 1354828, 6328548, 6679881, ...	13705481	Follow A005849 in OEIS
3	2, 8, 32, 54, 114, 414, 1400, 1850, 2848, 4874, 7268, 19290, 337590, 1183414, ...	1200000	Follow A006552 in OEIS
4th	(1), 3, 7, 33, 67, 223, 663, 912, 1383, 3777, 3972, 10669, 48375, ...	250000	Follow A007646 in OEIS
5	1242, 18390, ...	379575
6th	(1, 2), 91, 185, 387, 488, 747, 800, 9901, 10115, 12043, 13118, 30981, 51496, ...	200,000	Follow A242176 in OEIS
7th	34, 1980, 9898, ...	255681	Follow A242177 in OEIS
8th	(5), 17, 23, 1911, 20855, 35945, 42816, ..., 749130, ...	166666	Follow A242178 in OEIS
9	(2), 12382, 27608, 31330, 117852, ...	222431	Follow A265013 in OEIS
10	(1, 3), 9, 21, 363, 2161, 4839, 49521, 105994, 207777, ...	270026	Follow A007647 in OEIS
11	10, ...	600000
12	(1, 8), 247, 3610, 4775, 19789, 187895, ...	254519	Follow A242196 in OEIS
13	...	1000000
14th	(3, 5, 6, 9), 33, 45, 243, 252, 1798, 2429, 5686, 12509, 42545, ...	246922	Follow A242197 in OEIS
15th	(8), 14, 44, 154, 274, 694, 17426, 59430, ...	136149	Follow A242198 in OEIS
16	(1, 3), 55, 81, 223, 1227, 3012, 3301, ...	125000	Follow A242199 in OEIS
17th	19650, 236418, ...	281261
18th	(1, 3), 21, 23, 842, 1683, 3401, 16839, 49963, 60239, 150940, 155928, ...	203597	Follow A007648 in OEIS
19th	6460, ...	305777
20th	(3), 6207, 8076, 22356, 151456, ...	219976
21st	(2, 8), 26, 67100, ...	274099
22nd	(1, 15), 189, 814, 19909, 72207, ...	137649
23	4330, 89350, ...	177567
24	(2, 8), 368, ...	134188
25th	5610444, ...	500000
26th	117, 3143, 3886, 7763, 64020, 88900, ...	147626
27	(2), 56, 23454,…, 259738,…	215413
28	(1), 48, 468, 2655, 3741, 49930, ...	200618
29	...	500000
30th	(1, 2, 3, 7, 14, 17), 39, 79, 87, 99, 128, 169, 221, 252, 307, 3646, 6115, 19617, 49718, ...	101757

The largest generalized Cullen prime known to date is . It has 3,921,539 jobs and was discovered on September 3, 2019 by Tom Greer, a participant in the PrimeGrid Internet project. ${\displaystyle 2805222\cdot 25^{2805222}+1=2805222\cdot 5^{5610444}+1}$

See also

• Woodall number

literature

• J. Cullen: Question 15897 , Educ. Times, (December 1905) 534.

Web links

• Cullen Numbers
• Woodall Numbers

Individual evidence

1. ↑ PrimeGrid's Cullen Prime Search, 6679881 · 2 ^ 6679881 + 1. PrimeGrid, accessed November 2, 2016 . 
2. ↑ Chris K. Caldwell: The Top Twenty: Cullen Primes. Prime Pages, accessed April 26, 2018 . 
3. ↑ ^{a b} Weisstein, Eric W .: Cullen Number. MathWorld, accessed May 1, 2016 . 
4. ↑ ^{a b c d e f} Chris K. Caldwell: Cullen Prime. The Prime Glossary, accessed May 1, 2016 . 
5. ↑ Generalized Cullen primes nb ⁿ +1. Retrieved May 1, 2016 (List of Generalized Cullen Prime Numbers with base 3 to 100). 
6. ↑ List of generalized Cullen prime numbers with base 101 to 10000. Accessed May 1, 2016 . 
7. ↑ Chris K. Caldwell: The Largest Known Primes! 2805222 5 ^ 5610444 + 1. Prime Pages, accessed September 6, 2019 . 
8. ↑ Chris K. Caldwell: The Top Twenty: Generalized Cullen. Prime Pages, accessed September 6, 2019 .