Open problem, Arithmetic and number theory, posed 1640

Finiteness of Fermat primes

Open

It is conjectured that only finitely many Fermat numbers $F_n = 2^{2^n}+1$ are prime — in fact that the five known Fermat primes $F_0=3$ , $F_1=5$ , $F_2=17$ , $F_3=257$ , $F_4=65537$ are the only ones — even though Fermat himself originally believed every $F_n$ is prime.

Posed by, 1640Pierre de Fermat

Research frontier as of 2026

As of 2026, $F_5$ through $F_{32}$ have all been proven composite (via Pépin's test or an explicit factor), and no Fermat prime beyond $F_4$ has ever been found despite extensive distributed searches. A standard heuristic — treating $F_n$ as 'prime' with probability comparable to a random number of the same size — predicts the expected number of further Fermat primes beyond $F_4$ is about $3\times 10^{-10}$ , essentially zero, but this is not a proof that none exists.

Best known results

$F_n$ is proven composite for every $5 \le n \le 32$ ; complete prime factorizations are known only for $n \le 11$ .
Heuristic expected count of Fermat primes beyond $F_4$ is roughly $3\times 10^{-10}$ , based on treating primality as independent random events with probability $\approx 1/\ln F_n$ .

Tools and where they stop

Tool	Achieved	Where it stops
Pépin's test	Gives a fast, deterministic compositeness/primality decision for any single specified Fermat number, however large.	It only ever settles individual, finitely many candidates; it cannot rule out a prime appearing arbitrarily far out among untested indices.
Distributed factor search (PrimeGrid, Proth Search)	Finds explicit prime factors of enormous composite Fermat numbers, including some with index in the millions, confirming they are composite without a full factorization.	Finding no factor for a given $F_n$ up to some search depth neither proves it prime nor composite, and factor searches cannot cover the infinitude of untested indices.

Open questions

Is there a proof, even conditional, that only finitely many Fermat primes exist?
Is $F_{33}$ , the smallest Fermat number of unresolved status as of 2026, prime or composite?

References

Michal Křížek, Florian Luca, Lawrence Somer (2001). 17 Lectures on Fermat Numbers: From Number Theory to Geometry
Wilfrid Keller (2024). Fermat factoring status