Woodall number

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

Contents

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?

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, 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 is +1 and
W(3p 1)/2 if the Jacobi symbol 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.

The smallest value of n such that n × bn − 1 is prime for b = 1, 2, 3, ... 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 )

As of November 2021, the largest known generalized Woodall prime with base greater than 2 is 2740879 × 322740879  1. [7]

bNumbers n such that n × bn − 1 is prime [6] OEIS sequence
31, 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
41, 2, 3, 5, 8, 14, 23, 63, 107, 132, 428, 530, 1137, 1973, 2000, 7064, 20747, 79574, 113570, 293912, ..., 1993191, ... A086661
58, 14, 42, 384, 564, 4256, 6368, 21132, 27180, 96584, 349656, 545082, ... A059676
61, 2, 3, 19, 20, 24, 34, 77, 107, 114, 122, 165, 530, 1999, 4359, 11842, 12059, 13802, 22855, 41679, 58185, 145359, 249987, ... A059675
72, 18, 68, 84, 3812, 14838, 51582, ... A242200
81, 2, 7, 12, 25, 44, 219, 252, 507, 1155, 2259, 2972, 4584, 12422, 13905, 75606, ... A242201
910, 58, 264, 1568, 4198, 24500, ... A242202
102, 3, 8, 11, 15, 39, 60, 72, 77, 117, 183, 252, 396, 1745, 2843, 4665, 5364, ... A059671
112, 8, 252, 1184, 1308, ... A299374
121, 6, 43, 175, 821, 910, 1157, 13748, 27032, 71761, 229918, ... A299375
132, 6, 563528, ... A299376
141, 3, 7, 98, 104, 128, 180, 834, 1633, 8000, 28538, 46605, 131941, 147684, 433734, ... A299377
152, 10, 14, 2312, 16718, 26906, 27512, 41260, 45432, 162454, 217606, ... A299378
16167, 189, 639, ... A299379
172, 18, 20, 38, 68, 3122, 3488, 39500, ... A299380
181, 2, 6, 8, 10, 28, 30, 39, 45, 112, 348, 380, 458, 585, 17559, 38751, 43346, 46984, 92711, ... A299381
1912, 410, 33890, 91850, 146478, 189620, 280524, ... A299382
201, 18, 44, 60, 80, 123, 429, 1166, 2065, 8774, 35340, 42968, 50312, 210129, ... A299383

See also

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References

  1. Cunningham, A. J. C; Woodall, H. J. (1917), "Factorisation of and ", Messenger of Mathematics , 47: 1–38.
  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. 1 2 List of generalized Woodall primes base 3 to 10000
  7. "The Top Twenty: Generalized Woodall". primes.utm.edu. Retrieved 20 November 2021.

Further reading