Woodall number

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In number theory, a Woodall number (Wn) 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 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 Wn are prime are 2, 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, … (sequence A002234 in OEIS); the Woodall primes themselves begin with 7, 23, 383, 32212254719, … (sequence A050918 in 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 · 2n+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 December 2007, the largest known Woodall prime is 3752948 × 23752948 − 1.[3] It has 1,129,757 digits and was found by Matthew J. Thompson in 2007 in the distributed computing project PrimeGrid.

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]

Generalized Woodall number[edit]

A generalized Woodall number 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[4]

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, ... (sequence A240235 in OEIS)
b numbers n such that n × bn - 1 is prime 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, ... 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, ... A006553
4 1, 2, 3, 5, 8, 14, 23, 63, 107, 132, 428, 530, 1137, 1973, 2000, 7064, 20747, 79574, 113570, ... A086661
5 8, 14, 42, 384, 564, 4256, 6368, 21132, 27180, 96584, ... A059676
6 1, 2, 3, 19, 20, 24, 34, 77, 107, 114, 122, 165, 530, 1999, 4359, 11842, 12059, 13802, 22855, 41679, 58185, ... 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, ...
12 1, 6, 43, 175, 821, 910, 1157, 13748, 27032, 71761, 229918, ...
13 2, 6, 563528, ...
14 1, 3, 7, 98, 104, 128, 180, 834, 1633, 8000, 28538, 46605, 131941, 147684, ...
15 2, 10, 14, 2312, 16718, 26906, 27512, 41260, 45432, 162454, 217606, ...
16 167, 189, 639, ...
17 2, 18, 20, 38, 68, 3122, 3488, 39500, ...
18 1, 2, 6, 8, 10, 28, 30, 39, 45, 112, 348, 380, 458, 585, 17559, 38751, 43346, 46984, 92711, ...
19 12, 410, 33890, 91850, 146478, 189620, ...
20 1, 18, 44, 60, 80, 123, 429, 1166, 2065, 8774, 35340, 42968, 50312, ...
21 2, 18, 200, 282, 294, 1174, 2492, 4348, ...
22 2, 5, 140, 158, 263, 795, 992, ...
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, ...
30 1, 63, 331, 366, 1461, 3493, 4002, 5940, 13572, 34992, 182461, ...

See also[edit]

References[edit]

  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 .
  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: 938237*2^3752950-1", Chris Caldwell's The Largest Known Primes Database, retrieved December 22, 2009 
  4. ^ List of generalized Woodall primes

Further reading[edit]

External links[edit]