Are there infinitely many Newman–Shanks–Williams 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.

Let $S_0=1$, $S_1=7$ and $S_{n+2}=6S_{n+1}-S_n$ for $n\ge0$. Thus $S_n=\left((1+\sqrt2)^{2n+1}+(1-\sqrt2)^{2n+1}\right)/2$. These are the Newman--Shanks--Williams numbers; an NSW prime is a prime value $S_n$.

Statement recorded in the supplied source.

Are there infinitely many $n\ge0$ with $S_n$ prime? [S2]

Short English statement

Are infinitely many Newman–Shanks–Williams numbers prime?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400154 (AMR-093-0154). 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 Newman–Shanks–Williams 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, A002315: NSW numbers: a(n) = 6*a(n-1) - a(n-2); also a(n)^2 - 2*b(n)^2 = -1 with b(n) = A001653(n+1)., official data snapshot 693b5499c9d2 (2026) Name, definition and comments. https://raw.githubusercontent.com/oeis/oeisdata/693b5499c9d248f570dd2d979e8bd1d7eecf689b/seq/A002/A002315.seq.
• S4 OEIS Foundation and contributors, The On-Line Encyclopedia of Integer Sequences, A088165: NSW primes: NSW numbers that are also prime., official data snapshot 693b5499c9d2 (2026) Name, definition and comments. https://raw.githubusercontent.com/oeis/oeisdata/693b5499c9d248f570dd2d979e8bd1d7eecf689b/seq/A088/A088165.seq.