Math-4210 · mathematics · Number theory

Are there infinitely many Lucas primes

Status Open · unsolvedmath.comLLM status Open unverified

Statement

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.

Define the Lucas numbers by $L_0=2$, $L_1=1$ and $L_{n+2}=L_{n+1}+L_n$ for $n\ge0$. A Lucas prime is a prime value $L_n$.

Statement recorded in the supplied source.

Are there infinitely many $n\ge0$ such that $L_n$ is prime? [S2]

Short English statement

Do infinitely many Lucas numbers take prime values?

Sources

S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400152 (AMR-093-0152). 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 Lucas 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, A000032: Lucas numbers beginning at 2: L(n) = L(n-1) + L(n-2), L(0) = 2, L(1) = 1., official data snapshot 693b5499c9d2 (2026) Name, definition and comments. https://raw.githubusercontent.com/oeis/oeisdata/693b5499c9d248f570dd2d979e8bd1d7eecf689b/seq/A000/A000032.seq.
S4 OEIS Foundation and contributors, The On-Line Encyclopedia of Integer Sequences, A005479: Prime Lucas numbers (cf. A000032)., official data snapshot 693b5499c9d2 (2026) Name, definition and comments. https://raw.githubusercontent.com/oeis/oeisdata/693b5499c9d248f570dd2d979e8bd1d7eecf689b/seq/A005/A005479.seq.

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Record details

Subjects: Number theory
Collection: Top Open Problems
Mathematics rank: 4210
UnsolvedMath: Catalogue entry