For n≥ 0, define the Fermat numbers by F_n=2^2^n+1. Fermat prime finiteness conjecture. There are only finitely many Fermat primes, that is, only finitely many indices n for which F_n is prime. The heuristic is based on summing the approximate primality probabilities for the Fermat numbers; the conjecture remains open.

Fermat prime finiteness conjecture

Fermat conjectured in 1650 that every Fermat number was prime; Euler disproved this in 1732. The finiteness question remains open, with Hardy and Wright later supporting the expectation that only finitely many Fermat primes exist.

Known results

Euler, 1732: $F_5=641\cdot 6700417$ is composite.
Computations establish that $F_n$ is composite for $5\le n\le 32$ and for many further indices, including $n=18233954$ .
The only known Fermat primes are $F_0,F_1,F_2,F_3,F_4$ .
Probabilistic analyses, including Boklan–Conway and Caldwell, strongly favor finiteness but are not proofs.

2026 preprint

A 2026 preprint derives new primality and pseudoprimality constraints for Fermat numbers, but explicitly leaves open both the infinitude of Fermat primes and the infinitude of composite Fermat numbers; it does not claim to settle finiteness.

Current status (as of August 2026): The conjecture remains open; only five Fermat primes are known, many later Fermat numbers are computationally composite, and existing heuristics and the 2026 constraints provide no proof of finiteness.

Fermat prime finiteness conjecture

Primary source

Progress summary

Solutions 0