Are there infinitely many Kynea 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 integers $n\ge0$, define $K_n=(2^n+1)^2-2=4^n+2^{n+1}-1$. A Kynea prime is a prime value $K_n$.

Statement recorded in the supplied source.

Are there infinitely many $n\ge0$ for which $K_n$ is prime? [S2]

Short English statement

Are infinitely many Kynea numbers prime?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400151 (AMR-093-0151). 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 Kynea 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, A091513: Numbers k such that (2^k + 1)^2 - 2 = 4^k + 2^(k+1) - 1 is prime., official data snapshot 693b5499c9d2 (2026) Name, definition and comments. https://raw.githubusercontent.com/oeis/oeisdata/693b5499c9d248f570dd2d979e8bd1d7eecf689b/seq/A091/A091513.seq.