Is every Fermat number $2^{2^n} + 1$ composite for $n > 4$

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 $n\ge0$ the $n$th Fermat number is $F_n=2^{2^n}+1$. A composite integer is an integer greater than one that is not prime.

Statement recorded in the supplied source.

Is $F_n$ composite for every integer $n>4$? Equivalently, are $F_0,F_1,F_2,F_3,F_4$ the only Fermat primes? [S2]

Short English statement

Are all Fermat numbers after the first five composite?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400175 (AMR-093-0175). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, Is every Fermat number $2^{2^n} + 1$ composite for $n > 4$. 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, A000215: Fermat numbers: a(n) = 2^(2^n) + 1., official data snapshot 693b5499c9d2 (2026) Name, definition and comments. https://raw.githubusercontent.com/oeis/oeisdata/693b5499c9d248f570dd2d979e8bd1d7eecf689b/seq/A000/A000215.seq.