Wagstaff prime

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

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

The first few Wagstaff primes are:

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

As of October 2023, known 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, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937[2] (all known Wagstaff primes)

141079, 267017, 269987, 374321, 986191, 4031399, …, 13347311, 13372531, 15135397 (Wagstaff probable primes) (sequence A000978 in the OEIS)

In February 2010, Tony Reix discovered the Wagstaff probable prime:

${\displaystyle {\frac {2^{4031399}+1}{3}}}$

which has 1,213,572 digits and was the 3rd biggest probable prime ever found at this date.[3]

In September 2013, Ryan Propper announced the discovery of two additional Wagstaff probable primes:[4]

${\displaystyle {\frac {2^{13347311}+1}{3}}}$

and

${\displaystyle {\frac {2^{13372531}+1}{3}}}$

Each is a probable prime with slightly more than 4 million decimal digits. It is not currently known whether there are any exponents between 4031399 and 13347311 that produce Wagstaff probable primes.

In June 2021, Ryan Propper announced the discovery of the Wagstaff probable prime:[5]

${\displaystyle {\frac {2^{15135397}+1}{3}}}$

which is a probable prime with slightly more than 4.5 million decimal digits.

Primality has been proven or disproven for the values of p up to 138937. Those with p > 138937 are probable primes as of October 2023. The primality proof for p = 42737 was performed by François Morain in 2007 with a distributed ECPP implementation running on several networks of workstations for 743 GHz-days on an Opteron processor.[6] It was the third largest primality proof by ECPP from its discovery until March 2009.[7]

The Lucas–Lehmer–Riesel test can be used to identify Wagstaff PRPs.

It is natural to consider[8] more generally numbers of the form

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

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

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

these numbers are called "Wagstaff numbers base ${\displaystyle b}$", and sometimes considered[9] a case of the repunit numbers with negative base ${\displaystyle -b}$.

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

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

The least primes p such that ${\displaystyle Q(n,p)}$ 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 ${\displaystyle Q(b,prime(n))}$ 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)

• 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