Wagstaff prime

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In number theory, a Wagstaff prime is a prime number of the form

Wagstaff prime
Named afterSamuel S. Wagstaff, Jr.
Publication year1989[1]
Author of publicationBateman, P. T., Selfridge, J. L., Wagstaff Jr., S. S.
No. of known terms44
First terms3, 11, 43, 683
Largest known term(2138937+1)/3
OEIS index
  • A000979
  • Wagstaff primes: primes of form (2^p + 1)/3

where p is an odd prime. Wagstaff primes are named after the mathematician Samuel S. Wagstaff Jr.; the prime pages credit François Morain for naming them in a lecture at the Eurocrypt 1990 conference. Wagstaff primes appear in the New Mersenne conjecture and have applications in cryptography.

Examples

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The first three Wagstaff primes are 3, 11, and 43 because

 

Known Wagstaff primes

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The first few Wagstaff primes are:

3, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, ... (sequence A000979 in the OEIS)

Exponents which produce Wagstaff primes or probable primes are:

3, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, ... (sequence A000978 in the OEIS)

Generalizations

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It is natural to consider[2] more generally numbers of the form

 

where the base  . Since for   odd we have

 

these numbers are called "Wagstaff numbers base  ", and sometimes considered[3] a case of the repunit numbers with negative base  .

For some specific values of  , all   (with a possible exception for very small  ) are composite because of an "algebraic" factorization. Specifically, if   has the form of a perfect power with odd exponent (like 8, 27, 32, 64, 125, 128, 216, 243, 343, 512, 729, 1000, etc. (sequence A070265 in the OEIS)), then the fact that  , with   odd, is divisible by   shows that   is divisible by   in these special cases. Another case is  , with k a positive integer (like 4, 64, 324, 1024, 2500, 5184, etc. (sequence A141046 in the OEIS)), where we have the aurifeuillean factorization.

However, when   does not admit an algebraic factorization, it is conjectured that an infinite number of   values make   prime, notice all   are odd primes.

For  , the primes themselves have the following appearance: 9091, 909091, 909090909090909091, 909090909090909090909090909091, … (sequence A097209 in the OEIS), and these ns are: 5, 7, 19, 31, 53, 67, 293, 641, 2137, 3011, 268207, ... (sequence A001562 in the OEIS).

See Repunit#Repunit primes for the list of the generalized Wagstaff primes base  . (Generalized Wagstaff primes base   are generalized repunit primes base   with odd  )

The least primes p such that   is prime are (starts with n = 2, 0 if no such p exists)

3, 3, 3, 5, 3, 3, 0, 3, 5, 5, 5, 3, 7, 3, 3, 7, 3, 17, 5, 3, 3, 11, 7, 3, 11, 0, 3, 7, 139, 109, 0, 5, 3, 11, 31, 5, 5, 3, 53, 17, 3, 5, 7, 103, 7, 5, 5, 7, 1153, 3, 7, 21943, 7, 3, 37, 53, 3, 17, 3, 7, 11, 3, 0, 19, 7, 3, 757, 11, 3, 5, 3, ... (sequence A084742 in the OEIS)

The least bases b such that   is prime are (starts with n = 2)

2, 2, 2, 2, 2, 2, 2, 2, 7, 2, 16, 61, 2, 6, 10, 6, 2, 5, 46, 18, 2, 49, 16, 70, 2, 5, 6, 12, 92, 2, 48, 89, 30, 16, 147, 19, 19, 2, 16, 11, 289, 2, 12, 52, 2, 66, 9, 22, 5, 489, 69, 137, 16, 36, 96, 76, 117, 26, 3, ... (sequence A103795 in the OEIS)

References

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  1. ^ Bateman, P. T.; Selfridge, J. L.; Wagstaff, Jr., S. S. (1989). "The New Mersenne Conjecture". American Mathematical Monthly. 96: 125–128. doi:10.2307/2323195. JSTOR 2323195.
  2. ^ Dubner, H. and Granlund, T.: Primes of the Form (bn + 1)/(b + 1), Journal of Integer Sequences, Vol. 3 (2000)
  3. ^ Repunit, Wolfram MathWorld (Eric W. Weisstein)
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