Woodall Number
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Woodall Number

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

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

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

1, 7, 23, 63, 159, 383, 895, ... (sequence 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 Wn are prime are 2, 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, ... (sequence in the OEIS); the Woodall primes themselves begin with 7, 23, 383, 32212254719, ... (sequence 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 · 2n + 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, the largest known Woodall prime is 17016602 × 217016602 - 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 Wn to be prime, the index n cannot be congruent to 4 or 5 (modulo 6). Also, for a positive integer m, the Woodall number W2m may be prime only if 2m + 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 ${\displaystyle \left({\frac {2}{p}}\right)}$ is +1 and
W(3p - 1) / 2 if the Jacobi symbol ${\displaystyle \left({\frac {2}{p}}\right)}$ is -1.[]

## 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 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, the largest known generalized Woodall prime with base b greater than 1 is 17016602×217016602 - 1.

## References

1. ^ Cunningham, A. J. C; Woodall, H. J. (1917), "Factorisation of ${\displaystyle Q=(2^{q}\mp q)}$ and ${\displaystyle (q\cdot {2^{q}}\mp 1)}$", Messenger of Mathematics, 47: 1-38.
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. ^ 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 2020.
4. ^ "The Prime Database: 8508301*2^17016603-1", Chris Caldwell's The Largest Known Primes Database, retrieved 2018
5. ^ PrimeGrid, Announcement of 17016602*2^17016602 - 1 (PDF), retrieved 2018
6. ^ List of generalized Woodall primes base 3 to 10000