nLab Fermat number

A Fermat number is a number of the form

$$F_k = 2^{2^k}+1$$

for integers $k \geq 0$. A Fermat number that is prime is called a Fermat prime.

Pierre de Fermat conjectured (1640) that all $F_k$ are prime, but $F_5=4294967297$ is composite (641 is a factor, discovered by Euler in 1732), and in fact (as of 2014) only $F_k$ for $k=0,1,2,3,4$ are known to be prime. Basic questions remain open, such as:

• Are there infinitely many Fermat primes?
• Are there infinitely many composite Fermat numbers?

The Fermat numbers $F_k$ for $k=5,\ldots,32$ are known to be composite, though in some cases no explicit factors are known ($k=20,24$), or incomplete factorisations ($12 \leq k \leq 32$). Isolated other cases are known to be composite, for instance $k=3329780$.

There are heuristic (nonrigorous) arguments that there should be only finitely many Fermat primes, and that Fermat primes have positive density in the primes; it is impossible that both of these hold, since finite sets have zero density in a countable set like that of the primes. In any case, it is conjectured by various people that: there are only finitely many Fermat primes; there are only five Fermat primes; and that there are finitely many, but more than five (this last position is argued by Emil Artin).

Related entries

• prime number
• constructible polygon?
• Mersenne prime?

References

• Wikipedia
• OEIS?A000215
• Křížek, Michal; Luca, Florian; Somer, Lawrence 17 lectures on Fermat numbers. From number theory to geometry. With a foreword by Alena Šolcová. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 9. Springer-Verlag, New York, 2001. xxiv+257 pp. ISBN: 0-387-95332-9 doi:10.1007/978-0-387-21850-2

The following page lists heuristic arguments as to the finiteness of the Fermat primes, and also that there are only five such:

• The prime puzzles and conjectures connection, Conjecture 4. Fermat primes are finite

New factors of composite Fermat numbers are announced here:

• FermatSearch.org News