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In number theory, a Woodall number (W_n) is any natural number of the form

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

for some natural number n. The first few Woodall numbers are:

1, 7, 23, 63, 159, 383, 895, … (sequence A003261 in the OEIS).

History

Woodall numbers were first studied by Allan J. C. Cunningham and H. J. Woodall in 1917,^[1] inspired by James Cullen's earlier study of the similarly-defined Cullen numbers.

Woodall primes

Unsolved problem in mathematics:

Are there infinitely many Woodall primes?

(more unsolved problems in mathematics)

Woodall numbers that are also prime numbers are called Woodall primes; the first few exponents n for which the corresponding Woodall numbers W_n are prime are 2, 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, … (sequence A002234 in the OEIS); the Woodall primes themselves begin with 7, 23, 383, 32212254719, … (sequence A050918 in the OEIS).

In 1976 Christopher Hooley showed that almost all Cullen numbers are composite.^[2] In October 1995, Wilfred Keller published a paper discussing several new Cullen primes and the efforts made to factorise other Cullen and Woodall numbers. Included in that paper is a personal communication to Keller from Hiromi Suyama, asserting that Hooley's method can be reformulated to show that it works for any sequence of numbers $n \cdot 2 n + a + b$ , where a and b are integers, and in particular, that Woodall numbers are almost all composites.^[3] It is an open problem on whether there are infinitely many Woodall primes. As of October 2018^[update], the largest known Woodall prime is 17016602 × 2^17016602 − 1.^[4] It has 5,122,515 digits and was found by Diego Bertolotti in March 2018 in the distributed computing project PrimeGrid.^[5]

Restrictions

Starting with W₄ = 63 and W₅ = 159, every sixth Woodall number is divisible by 3; thus, in order for W_n to be prime, the index n cannot be congruent to 4 or 5 (modulo 6). Also, for a positive integer m, the Woodall number W_2^m may be prime only if 2^m + m is prime. As of January 2019, the only known primes that are both Woodall primes and Mersenne primes are W₂ = M₃ = 7, and W₅₁₂ = M₅₂₁.

Divisibility properties

Like Cullen numbers, Woodall numbers have many divisibility properties. For example, if p is a prime number, then p divides

W_{(p + 1) / 2} if the Jacobi symbol

\left({\frac {2}{p}}\right)

is +1 and

W_{(3p − 1) / 2} if the Jacobi symbol

\left({\frac {2}{p}}\right)

is −1.^{[citation needed]}

Generalization

A generalized Woodall 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 Woodall prime.

Least n such that n × bⁿ - 1 is prime are^[6]

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, 2, 3, 2, 8, 60, 1, 2, 2, 10, 5, 2, 7, 182, 1, 17782, 3, ... (sequence A240235 in the OEIS)

b	numbers n such that n × bⁿ - 1 is prime (these n are checked up to 350000)	OEIS sequence
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 primes plus 1)	A008864
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, ...	A002234
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, ...	A006553
4	1, 2, 3, 5, 8, 14, 23, 63, 107, 132, 428, 530, 1137, 1973, 2000, 7064, 20747, 79574, 113570, 293912, ..., 1993191, ...	A086661
5	8, 14, 42, 384, 564, 4256, 6368, 21132, 27180, 96584, 349656, 545082, ...	A059676
6	1, 2, 3, 19, 20, 24, 34, 77, 107, 114, 122, 165, 530, 1999, 4359, 11842, 12059, 13802, 22855, 41679, 58185, 145359, 249987, ...	A059675
7	2, 18, 68, 84, 3812, 14838, 51582, ...	A242200
8	1, 2, 7, 12, 25, 44, 219, 252, 507, 1155, 2259, 2972, 4584, 12422, 13905, 75606, ...	A242201
9	10, 58, 264, 1568, 4198, 24500, ...	A242202
10	2, 3, 8, 11, 15, 39, 60, 72, 77, 117, 183, 252, 396, 1745, 2843, 4665, 5364, ...	A059671
11	2, 8, 252, 1184, 1308, ...	A299374
12	1, 6, 43, 175, 821, 910, 1157, 13748, 27032, 71761, 229918, ...	A299375
13	2, 6, 563528, ...	A299376
14	1, 3, 7, 98, 104, 128, 180, 834, 1633, 8000, 28538, 46605, 131941, 147684, 433734, ...	A299377
15	2, 10, 14, 2312, 16718, 26906, 27512, 41260, 45432, 162454, 217606, ...	A299378
16	167, 189, 639, ...	A299379
17	2, 18, 20, 38, 68, 3122, 3488, 39500, ...	A299380
18	1, 2, 6, 8, 10, 28, 30, 39, 45, 112, 348, 380, 458, 585, 17559, 38751, 43346, 46984, 92711, ...	A299381
19	12, 410, 33890, 91850, 146478, 189620, 280524, ...	A299382
20	1, 18, 44, 60, 80, 123, 429, 1166, 2065, 8774, 35340, 42968, 50312, 210129, ...	A299383
21	2, 18, 200, 282, 294, 1174, 2492, 4348, ...
22	2, 5, 140, 158, 263, 795, 992, 341351, ...
23	29028, ...
24	1, 2, 5, 12, 124, 1483, 22075, 29673, 64593, ...
25	2, 68, 104, 450, ...
26	3, 8, 79, 132, 243, 373, 720, 1818, 11904, 134778, ...
27	10, 18, 20, 2420, 6638, 11368, 14040, 103444, ...
28	2, 5, 6, 12, 20, 47, 71, 624, 1149, 2399, 8048, 30650, 39161, ...
29	26850, 237438, 272970, ...
30	1, 63, 331, 366, 1461, 3493, 4002, 5940, 13572, 34992, 182461, 201038, ...

As of October 2018^[update], the largest known generalized Woodall prime is 17016602×2^17016602 − 1.

Dual form

If we let n take negative values, and choose the numerator of the absolute value of these numbers, then we get

(−n)×b⁻ⁿ−1 = −(bⁿ+n)/(bⁿ)

and we choose the number bⁿ+n (we assume that n is coprime to bⁿ)

For b = 2, this number is prime for n =

1, 3, 5, 9, 15, 39, 75, 81, 89, 317, 701, 735, 1311, 1881, 3201, 3225, 11795, 88071, 204129, 678561, ... (sequence A052007 in the OEIS)

For b = 3, this number is prime for n =

2, 8, 34, 1532, 18248, ... (sequence A057900 in the OEIS)

Smallest n such that this number is prime for b = 1, 2, 3, ... are

2, 1, 1, 2, 1, 7954, 1, 34, 101, 2, 1, ... (sequence A093324 in the OEIS)

References

^ Cunningham, A. J. C; Woodall, H. J. (1917), "Factorisation of $Q=(2^{q}\mp q)$ and $(q\cdot {2^{q}}\mp 1)$ ", Messenger of Mathematics, 47: 1–38.
^ Everest, Graham; van der Poorten, Alf; Shparlinski, Igor; Ward, Thomas (2003). Recurrence sequences. Mathematical Surveys and Monographs. Vol. 104. Providence, RI: American Mathematical Society. p. 94. ISBN 0-8218-3387-1. Zbl 1033.11006.
^ Keller, Wilfrid (January 1995). "New Cullen primes". Mathematics of Computation. 64 (212): 1739. doi:10.1090/S0025-5718-1995-1308456-3. ISSN 0025-5718. Keller, Wilfrid (December 2013). "Wilfrid Keller". www.fermatsearch.org. Hamburg. Archived from the original on February 28, 2020. Retrieved October 1, 2020.
^ "The Prime Database: 8508301*2^17016603-1", Chris Caldwell's The Largest Known Primes Database, retrieved March 24, 2018
^ PrimeGrid, Announcement of 17016602*2^17016602 - 1 (PDF), retrieved April 1, 2018
^ List of generalized Woodall primes base 3 to 10000

External links

Chris Caldwell, The Prime Glossary: Woodall number, and The Top Twenty: Woodall, and The Top Twenty: Generalized Woodall, at The Prime Pages.
Weisstein, Eric W. "Woodall number". MathWorld.
Steven Harvey, List of Generalized Woodall primes.
Paul Leyland, Generalized Cullen and Woodall Numbers
Woodall number in Prime wiki

[1] Cunningham, A. J. C; Woodall, H. J. (1917), "Factorisation of $Q=(2^{q}\mp q)$ and $(q\cdot {2^{q}}\mp 1)$ ", Messenger of Mathematics, 47: 1–38.

[EPSW94-2] Everest, Graham; van der Poorten, Alf; Shparlinski, Igor; Ward, Thomas (2003). Recurrence sequences. Mathematical Surveys and Monographs. Vol. 104. Providence, RI: American Mathematical Society. p. 94. ISBN 0-8218-3387-1. Zbl 1033.11006.

[3] Keller, Wilfrid (January 1995). "New Cullen primes". Mathematics of Computation. 64 (212): 1739. doi:10.1090/S0025-5718-1995-1308456-3. ISSN 0025-5718. Keller, Wilfrid (December 2013). "Wilfrid Keller". www.fermatsearch.org. Hamburg. Archived from the original on February 28, 2020. Retrieved October 1, 2020.

[4] "The Prime Database: 8508301*2^17016603-1", Chris Caldwell's The Largest Known Primes Database, retrieved March 24, 2018

[5] PrimeGrid, Announcement of 17016602*2^17016602 - 1 (PDF), retrieved April 1, 2018

[6] List of generalized Woodall primes base 3 to 10000

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@@ Line 171: / Line 171: @@
 ==Dual form==
 If we let ''n'' take negative values, and choose the [[numerator]] of the [[absolute value]] of these numbers, then we get
 :(−''n'')×''b''<sup>−''n''</sup>−1 = −(''b''<sup>''n''</sup>+''n'')/(''b''<sup>''n''</sup>)