Wagstaff prime

Wagstaff prime
Named after	Samuel S. Wagstaff, Jr.
Publication year	1989
Author of publication	Bateman, P. T., Selfridge, J. L., Wagstaff Jr., S. S.
No. of known terms	44
First terms	3, 11, 43, 683
Largest known term	(2138937+1)/3
OEIS index	A000979; Wagstaff primes: primes of form (2^p + 1)/3;

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

{{2^{p}+1} \over 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[edit]

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

{\begin{aligned}3&={2^{3}+1 \over 3},\\[5pt]11&={2^{5}+1 \over 3},\\[5pt]43&={2^{7}+1 \over 3}.\end{aligned}}

Known Wagstaff primes[edit]

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[edit]

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

Q(b,n)={\frac {b^{n}+1}{b+1}}

where the base $b\geq 2$ . Since for $n$ odd we have

{\frac {b^{n}+1}{b+1}}={\frac {(-b)^{n}-1}{(-b)-1}}=R_{n}(-b)

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

For some specific values of $b$ , all $Q(b,n)$ (with a possible exception for very small $n$ ) are composite because of an "algebraic" factorization. Specifically, if $b$ 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 $x^{m}+1$ , with $m$ odd, is divisible by $x+1$ shows that $Q(a^{m},n)$ is divisible by $a^{n}+1$ in these special cases. Another case is $b=4k^{4}$ , 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 $b$ does not admit an algebraic factorization, it is conjectured that an infinite number of $n$ values make $Q(b,n)$ prime, notice all $n$ are odd primes.

For $b=10$ , 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 $b$ . (Generalized Wagstaff primes base $b$ are generalized repunit primes base $-b$ with odd $n$ )

The least primes p such that $Q(n,p)$ 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 $Q(b,prime(n))$ 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[edit]

^ 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.
^ Dubner, H. and Granlund, T.: Primes of the Form (bⁿ + 1)/(b + 1), Journal of Integer Sequences, Vol. 3 (2000)
^ Repunit, Wolfram MathWorld (Eric W. Weisstein)

External links[edit]

John Renze and Eric W. Weisstein. "Wagstaff prime". MathWorld.
Chris Caldwell, The Top Twenty: Wagstaff at The Prime Pages.
Renaud Lifchitz, "An efficient probable prime test for numbers of the form (2^p + 1)/3".
Tony Reix, "Three conjectures about primality testing for Mersenne, Wagstaff and Fermat numbers based on cycles of the Digraph under x² − 2 modulo a prime".
List of repunits in base -50 to 50
List of Wagstaff primes base 2 to 160

[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)

[1]

[2]

[3]

v t e Prime number classes
By formula	Fermat ( $2 2 n + 1$ ) Mersenne ( $2 p - 1$ ) Double Mersenne ( $2 2 p -1 - 1$ ) Wagstaff $(2 p + 1)/3$ Proth ( $k \cdot2 n + 1$ ) Factorial ( $n! \pm 1$ ) Primorial ( $p n # \pm 1$ ) Euclid ( $p n # + 1$ ) Pythagorean ( $4 n + 1$ ) Pierpont ( $2 m \cdot3 n + 1$ ) Quartan ( $x 4 + y 4$ ) Solinas ( $2 m \pm 2 n \pm 1$ ) Cullen ( $n \cdot2 n + 1$ ) Woodall ( $n \cdot2 n - 1$ ) Cuban ( $x 3 - y 3)/(x - y$ ) Leyland ( $x y + y x$ ) Thabit ( $3\cdot2 n - 1$ ) Williams ( $(b -1)\cdot b n - 1$ ) Mills ( $⌊ A 3 n ⌋$ )
By integer sequence	Fibonacci Lucas Pell Newman–Shanks–Williams Perrin
By property	Wieferich (pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient Unique
Base-dependent	Palindromic Emirp Repunit $(10 n - 1)/9$ Permutable Circular Truncatable Minimal Delicate Primeval Full reptend Unique Happy Self Smarandache–Wellin Strobogrammatic Dihedral Tetradic
Patterns	Twin ( $p, p + 2$ ) Bi-twin chain ( $n \pm 1, 2 n \pm 1, 4 n \pm 1, \dots$ ) Triplet ( $p, p + 2 or p + 4, p + 6$ ) Quadruplet ( $p, p + 2, p + 6, p + 8$ ) k-tuple Cousin ( $p, p + 4$ ) Sexy ( $p, p + 6$ ) Chen Sophie Germain/Safe ( $p, 2 p + 1$ ) Cunningham ( $p, 2 p \pm 1, 4 p \pm 3, 8 p \pm 7, ...$ ) Arithmetic progression ( $p + a\cdotn, n = 0, 1, 2, 3, ...$ ) Balanced ( $consecutive p - n, p, p + n$ )
By size	Mega (1,000,000+ digits) Largest known list
Complex numbers	Eisenstein prime Gaussian prime
Composite numbers	Pseudoprime Catalan Elliptic Euler Euler–Jacobi Fermat Frobenius Lucas Perrin Somer–Lucas Strong Carmichael number Almost prime Semiprime Sphenic number Interprime Pernicious
Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap
First 60 primes	2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
List of prime numbers