New Mersenne conjecture

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 positive integer $p$, consider these three conditions: (i) for some integer $k\ge1$, $p$ equals one of $2^k-1$, $2^k+1$, $4^k-3$, $4^k+3$; (ii) $2^p-1$ is prime; (iii) $(2^p+1)/3$ is prime. The quotient in (iii) is an integer because $p$ is odd. The variable $p$ is not assumed prime in advance.

Statement recorded in the supplied source.

Whenever any two of conditions (i), (ii) and (iii) hold for such $p$, does the remaining condition also hold? [S2]

Short English statement

For every odd positive exponent, do any two of these special-exponent, Mersenne-prime and Wagstaff-prime properties force the third?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400133 (AMR-093-0133). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, New Mersenne conjecture. 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, A000668: Mersenne primes (primes of the form 2^n - 1)., official data snapshot 693b5499c9d2 (2026) Name, definition and comments. https://raw.githubusercontent.com/oeis/oeisdata/693b5499c9d248f570dd2d979e8bd1d7eecf689b/seq/A000/A000668.seq.
• S4 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.