Woodall number

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[edit]

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[edit]

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] Hooley's proof was reworked by Hiromi Suyama to show that it works for any sequence of numbers n · 2^n+a + b where a and b are integers, and in particular also for Woodall numbers. Nonetheless, it is conjectured that there are infinitely many Woodall primes.^{[citation needed]} As of October 2018^[update], the largest known Woodall prime is 17016602 × 2^17016602 − 1.^[3] It has 5,122,515 digits and was found by Diego Bertolotti in March 2018 in the distributed computing project PrimeGrid.^[4]

Restrictions[edit]

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[edit]

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[edit]

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^[5]

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.

References[edit]

^ 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. 104. Providence, RI: American Mathematical Society. p. 94. ISBN 0-8218-3387-1. Zbl 1033.11006.
^ "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[edit]

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

[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. 104. Providence, RI: American Mathematical Society. p. 94. ISBN 0-8218-3387-1. Zbl 1033.11006.

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

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

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

[1]

[2]

[3]

[4]

[5]

v t e Prime number classes
By formula	Fermat ( $2 2 n + 1$ ) Mersenne ( $2 p - 1$ ) Double Mersenne ( $2 2 p -1 - 1$ ) Wagstaff $(2 p + 1)/3$ Proth ( $k \cdot2 n + 1$ ) Factorial ( $n! \pm 1$ ) Primorial ( $p n # \pm 1$ ) Euclid ( $p n # + 1$ ) Pythagorean ( $4 n + 1$ ) Pierpont ( $2 m \cdot3 n + 1$ ) Quartan ( $x 4 + y 4$ ) Solinas ( $2 m \pm 2 n \pm 1$ ) Cullen ( $n \cdot2 n + 1$ ) Woodall ( $n \cdot2 n - 1$ ) Cuban ( $x 3 - y 3)/(x - y$ ) Carol $(2 n - 1) 2 - 2$ Kynea $(2 n + 1) 2 - 2$ Leyland ( $x y + y x$ ) Thabit ( $3\cdot2 n - 1$ ) Williams ( $(b -1)\cdot b n - 1$ ) Mills ( $⌊ A 3 n ⌋$ )
By integer sequence	Fibonacci Lucas Pell Newman–Shanks–Williams Perrin Partitions Bell Motzkin
By property	Wieferich (pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient
Base-dependent	Happy Dihedral Palindromic Emirp Repunit $(10 n - 1)/9$ Permutable Circular Truncatable Strobogrammatic Minimal Weakly Full reptend Unique Primeval Self Smarandache–Wellin Tetradic
Patterns	Twin ( $p, p + 2$ ) Bi-twin chain ( $n - 1, n + 1, 2 n - 1, 2 n + 1, \dots$ ) Triplet ( $p, p + 2 or p + 4, p + 6$ ) Quadruplet ( $p, p + 2, p + 6, p + 8$ ) k−Tuple Cousin ( $p, p + 4$ ) Sexy ( $p, p + 6$ ) Chen Sophie Germain ( $p, 2 p + 1$ ) Cunningham ( $p, 2 p \pm 1, 4 p \pm 3, 8 p \pm 7, ...$ ) Safe ( $p, (p - 1)/2$ ) Arithmetic progression ( $p + a\cdotn, n = 0, 1, 2, 3, ...$ ) Balanced ( $consecutive p - n, p, p + n$ )
By size	Titanic (1,000+ digits) Gigantic (10,000+ digits) Mega (1,000,000+ digits) Largest known
Complex numbers	Eisenstein prime Gaussian prime
Composite numbers	Pseudoprime Catalan Elliptic Euler Euler–Jacobi Fermat Frobenius Lucas Somer–Lucas Strong Carmichael number Almost prime Semiprime Interprime Pernicious
Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap
First 60 primes	2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
List of prime numbers

Woodall number

History[edit]

Woodall primes[edit]

Restrictions[edit]

Divisibility properties[edit]

Generalization[edit]

See also[edit]

References[edit]

Further reading[edit]

External links[edit]

Navigation menu

Personal tools

Namespaces

Variants

Views

More

Search

Navigation

Interaction

Tools

Print/export

Languages