Are there infinitely many Wagstaff primes

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For an odd prime $p$, the integer $W_p=(2^p+1)/3$ is a Wagstaff number; a prime value is a Wagstaff prime. Equivalently, one may use all positive integer exponents $n$ for which $(2^n+1)/3$ is an integer prime: a prime value forces $n$ to be an odd prime.

Statement recorded in the supplied source.

Are there infinitely many odd primes $p$ for which $W_p$ is prime? [S2]

Short English statement

Are infinitely many numbers two to an odd prime power, plus one and divided by three, prime?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400163 (AMR-093-0163). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, Are there infinitely many Wagstaff primes. https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory.
• S3 OEIS Foundation and contributors, The On-Line Encyclopedia of Integer Sequences, A000978: Wagstaff numbers: numbers k such that (2^k + 1)/3 is prime., official data snapshot 693b5499c9d2 (2026) Name, definition and comments. https://raw.githubusercontent.com/oeis/oeisdata/693b5499c9d248f570dd2d979e8bd1d7eecf689b/seq/A000/A000978.seq.