Woodall number

A Woodall number is a natural number of the form:

${\displaystyle W_{n}=n\cdot 2^{n}-1}$

for a natural number . The first Woodall numbers are: ${\displaystyle n\geq 1}$

1, 7, 23, 63, 159, 383, 895, 2047, 10239, 22527, 49151, 106495, 229375, 491519, 1048575, 2228223, 4718591, 9961471, 20971519, 44040191, ... (sequence A003261 in OEIS )

history

Woodall numbers were first described by Allan JC Cunningham and HJ Woodall in 1917. Both were inspired by James Cullen , who defined a similar sequence of numbers: the Cullen numbers .

Similar consequences

The Cullen numbers are defined by:

${\displaystyle C_{n}=n\cdot 2^{n}+1,n\in \mathbb {N} }$

As a result:

${\displaystyle C_{n}-W_{n}=2}$.

Because of this similarity, Woodall numbers are also referred to as 2nd order Cullen numbers .

Woodall primes

A Woodall number that is also prime is called a Woodall prime number . The first exponents for which Woodall numbers represent such Woodall prime numbers are: ${\displaystyle n}$

${\displaystyle n}$= 2, 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, 462, 512, 751, 822, 5312, 7755, 9531, 12379, 15822, 18885, 22971, 23005, 98726, 143018 , 151023, 667071, 1195203, 1268979, 1467763, 2013992, 2367906, 3752948, 17016602, ... (sequence A002234 in OEIS )

${\displaystyle W_{n}}$= 7, 23, 383, 32212254719, 2833419889721787128217599, ... (sequence A050918 in OEIS )

In particular, the larger Woodall prime numbers were found by the BOINC project PrimeGrid .

The largest Woodall prime to date was calculated on March 22, 2018 and is:

${\displaystyle W_{17016602}=17016602\cdot 2^{17016602}-1=8508301\cdot 2^{17016603}-1}$

That number has 5,122,515 digits and was discovered by the Italian Diego Bertolotti, a participant in the Internet project PrimeGrid.

It is known that there are no more prime Woodall numbers up to . It is assumed , however , that there are an infinite number of Woodall prime numbers. ${\displaystyle n<14508061}$

Properties of Woodall numbers

• Almost all Woodall numbers are composite numbers (proven by Christopher Hooley in 1976).
• The prime number divides Woodall's number when is Jacobi symbol . ${\displaystyle p}$ ${\displaystyle W_{\frac {p+1}{2}}}$ ${\displaystyle \left({\frac {2}{p}}\right)=+1}$
• The prime number divides Woodall's number when is Jacobi symbol . ${\displaystyle p}$ ${\displaystyle W_{\frac {3p-1}{2}}}$ ${\displaystyle \left({\frac {2}{p}}\right)=-1}$
• The following applies:

${\displaystyle W_{4}=4\cdot 2^{4}-1=63}$and are both divisible by three. ${\displaystyle W_{5}=5\cdot 2^{5}-1=159}$

Every sixth Woodall number is also divisible by. Thus, a Woodall prime number is only possible if the index is not a multiple of 4 or 5 ( modulo 6). ${\displaystyle W_{n}}$ ${\displaystyle 3}$ ${\displaystyle W_{n}}$ ${\displaystyle n}$

• The only two known prime numbers that represent Woodall prime numbers and Mersenne prime numbers at the same time are (as of May 2019):

${\displaystyle W_{2}=M_{3}=7}$ and ${\displaystyle W_{512}=M_{521}}$

Generalized Woodall numbers

Numbers of the form with are called generalized Woodall numbers . ${\displaystyle n\cdot b^{n}-1}$ ${\displaystyle n+2>b}$

If this number is a prime number, it is called a generalized Woodall prime number .

The condition is necessary because without this condition every prime number would be a generalized Woodall prime because would be. ${\displaystyle n+2>b}$ ${\displaystyle p}$ ${\displaystyle p=1\cdot (p+1)^{1}-1}$

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

3, 2, 1, 1, 8, 1, 2, 1, 10, 2, 2, 1, 2, 1, 2, 167, 2, 1, 12, 1, 2, 2, 29028, 1, 2, 3, 10, 2, 26850, 1, 8, 1, 42, 2, 6, 2, 24, 1, 2, 3, 2, 1, 2, 1, 2, 2, 140, 1, 2, 2, 22, 2, 8, 1, 2064, 2, 468, 6, 2, 1, 362, 1, 2, 2, 6, 3, 26, 1, 2, 3, 20, 1, 2, 1, 28, 2, 38, 5, 3024, 1, 2, 81, 858, 1, ... (Follow A240235 in OEIS )

The following is a listing of the first generalized Woodall primes for bases between 1 and 30. These have been studied up to at least 200,000. If the condition does not apply, but the number is still prime, it is put in brackets: ${\displaystyle b}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n+2>b}$ ${\displaystyle n\cdot b^{n}-1}$

${\displaystyle b}$	${\displaystyle n}$so that is prime ${\displaystyle n\cdot b^{n}-1}$	examined up	OEIS episode
1	3, 4, 6, 8, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 54, 60, 62, 68, 72, 74, 80, 84, 90, 98, 102, 104, 108, 110, 114, 128, 132, 138, 140, 150, 152, 158, 164, 168, 174, 180, 182, 192, 194, 198, 200, 212, 224, 228, 230, 234, 240, 242, 252, 258, 264, 270, 272, 278, 282, 284, 294, ... (all prime numbers plus 1)	all prime numbers	Follow A008864 in OEIS
2	2, 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, 462, 512, 751, 822, 5312, 7755, 9531, 12379, 15822, 18885, 22971, 23005, 98726, 143018, 151023, 667071, 1195203, 1268979, 1467763, 2013992, 2367906, 3752948, 17016602, ...	14508061	Follow A002234 in OEIS
3	(1), 2, 6, 10, 18, 40, 46, 86, 118, 170, 1172, 1698, 1810, 2268, 4338, 18362, 72662, 88392, 94110, 161538, 168660, 292340, 401208, 560750, 1035092, ...	1058000	Follow A006553 in OEIS
4th	(1, 2), 3, 5, 8, 14, 23, 63, 107, 132, 428, 530, 1137, 1973, 2000, 7064, 20747, 79574, 113570, 293912, ..., 1993191, ...	1000000	Follow A086661 in OEIS
5	8, 14, 42, 384, 564, 4256, 6368, 21132, 27180, 96584, 349656, 545082, ...	1000000	Follow A059676 in OEIS
6th	(1, 2, 3), 19, 20, 24, 34, 77, 107, 114, 122, 165, 530, 1999, 4359, 11842, 12059, 13802, 22855, 41679, 58185, 145359, 249987, ...	876000	Follow A059675 in OEIS
7th	(2), 18, 68, 84, 3812, 14838, 51582, ...	350000	Follow A242200 in OEIS
8th	(1, 2), 7, 12, 25, 44, 219, 252, 507, 1155, 2259, 2972, 4584, 12422, 13905, 75606, ...	513000	Follow A242201 in OEIS
9	10, 58, 264, 1568, 4198, 24500, ...	975000	Follow A242202 in OEIS
10	(2, 3, 8), 11, 15, 39, 60, 72, 77, 117, 183, 252, 396, 1745, 2843, 4665, 5364, ...	500000	Follow A059671 in OEIS
11	(2, 8), 252, 1184, 1308, ...	500000	Follow A299374 in OEIS
12	(1, 6), 43, 175, 821, 910, 1157, 13748, 27032, 71761, 229918, ...	500000	Follow A299375 in OEIS
13	(2, 6), 563528, ...	570008	Follow A299376 in OEIS
14th	(1, 3, 7), 98, 104, 128, 180, 834, 1633, 8000, 28538, 46605, 131941, 147684, 433734, ...	500000	Follow A299377 in OEIS
15th	(2, 10), 14, 2312, 16718, 26906, 27512, 41260, 45432, 162454, 217606, ...	500000	Follow A299378 in OEIS
16	167, 189, 639, ...	500000	Follow A299379 in OEIS
17th	(2), 18, 20, 38, 68, 3122, 3488, 39500, ...	400,000	Follow A299380 in OEIS
18th	(1, 2, 6, 8, 10), 28, 30, 39, 45, 112, 348, 380, 458, 585, 17559, 38751, 43346, 46984, 92711, ...	400,000	Follow A299381 in OEIS
19th	(12), 410, 33890, 91850, 146478, 189620, 280524, ...	400,000	Follow A299382 in OEIS
20th	(1, 18), 44, 60, 80, 123, 429, 1166, 2065, 8774, 35340, 42968, 50312, 210129, ...	250000	Follow A299383 in OEIS
21st	(2, 18), 200, 282, 294, 1174, 2492, 4348, ...	200,000
22nd	(2, 5), 140, 158, 263, 795, 992, ...	200,000
23	29028, ...	200,000
24	(1, 2, 5, 12), 124, 1483, 22075, 29673, 64593, ...	200,000
25th	(2), 68, 104, 450, ...	500000
26th	(3, 8), 79, 132, 243, 373, 720, 1818, 11904, 134778, ...	200,000
27	(10, 18, 20), 2420, 6638, 11368, 14040, 103444, ...	450000
28	(2, 5, 6, 12, 20), 47, 71, 624, 1149, 2399, 8048, 30650, 39161, ...	200,000
29	26850, ...	200,000
30th	(1), 63, 331, 366, 1461, 3493, 4002, 5940, 13572, 34992, 182461, ...	200,000

The largest known generalized Woodall prime to date is . It has 4,125,441 jobs and was discovered by Ryan Propper on October 26, 2019. ${\displaystyle 2740879\cdot 32^{2740879}-1=2740879\cdot 2^{13704395}-1}$

See also

• Cullen's number

literature

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

Web links

• Cullen prime
• Woodall prime

Individual evidence

1. ^ AJ C Cunningham, HJ Woodall: Factorization of and ${\displaystyle Q=(2^{q}\mp q)}$ ${\displaystyle (q\cdot {2^{q}}\mp 1)}$ . In: Messenger of Mathematics . 1917, p. 1 of 151 .
2. ↑ Eric W. Weisstein : Woodall Number. Retrieved May 25, 2019 . 
3. ↑ PrimeGrid's Woodall Prime Search, 17016602 · 2 ^17016602 - 1. PrimeGrid, accessed April 26, 2018 . 
4. ↑ Chris K. Caldwell: The Top Twenty: Woodall Primes. Prime Pages, accessed April 26, 2018 . 
5. ^ Weisstein, Eric W .: Woodall Number. MathWorld, accessed May 1, 2016 . 
6. ↑ ^{a b c d} Chris K. Caldwell: Woodall Prime. The Prime Glossary, accessed May 1, 2016 . 
7. ↑ Graham Everest, Alf van der Poorten, Igor Shparlinski, Thomas Ward: Recurrence sequences. Mathematical Surveys and Monographs. In: RI: American Mathematical Society . 2003, ISBN 0-8218-3387-1 , pp. 94 .
8. ↑ List of generalized Woodall prime numbers with base 3 to 10000. Accessed May 1, 2016 . 
9. ↑ Chris K. Caldwell: The Largest Known Primes! 2740879 32 ^2740879 - 1. Prime Pages, accessed January 15, 2020 . 
10. ↑ Chris K Caldwell: The Top Twenty: Generalized Woodall. Prime Pages, accessed January 15, 2020 .