In number theory, a Woodall number (Wn) is any natural number of the form
for some natural number n. The first few Woodall numbers are:
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.
Are there infinitely many Woodall primes?
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 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 · 2n + a + b, where a and b are integers, and in particular, that almost all Woodall numbers are composite. [3] It is an open problem whether there are infinitely many Woodall primes. As of October 2018 [update] , 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]
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.
Like Cullen numbers, Woodall numbers have many divisibility properties. For example, if p is a prime number, then p divides
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.
The smallest value of n such that n × bn − 1 is prime for b = 1, 2, 3, ... are [6]
As of November 2021 [update] , the largest known generalized Woodall prime with base greater than 2 is 2740879 × 322740879 − 1. [7]
b | Numbers n such that n × bn − 1 is prime [6] | OEIS sequence |
---|---|---|
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 |
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