Wagstaff prime

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
The first three Wagstaff primes are 3, 11, and 43 because
 * $$\begin{align}

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

Known Wagstaff primes
The first few Wagstaff primes are:
 * 3, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, ...

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, ...

Generalizations
It is natural to consider more generally numbers of the form
 * $$Q(b,n)=\frac{b^n+1}{b+1}$$

where the base $$b \ge 2$$. 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 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. ), 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. ), 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, …, and these ns are: 5, 7, 19, 31, 53, 67, 293, 641, 2137, 3011, 268207, ... .

See Repunit 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, ...

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, ...