nLab
Fermat number

A Fermat number is a number of the form

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

for integers k0k \geq 0. A Fermat number that is prime is called a Fermat prime.

Pierre de Fermat conjectured (1640) that all F kF_k are prime, but F 5=4294967297F_5=4294967297 is composite (641 is a factor, discovered by Euler in 1732), and in fact (as of 2014) only F kF_k for k=0,1,2,3,4k=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 kF_k for k=5,,32k=5,\ldots,32 are known to be composite, though in some cases no explicit factors are known (k=20,24k=20,24), or incomplete factorisations (12k3212 \leq k \leq 32). Isolated other cases are known to be composite, for instance k=3329780k=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).

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:

New factors of composite Fermat numbers are announced here:

  • FermatSearch.org News
Revised on May 1, 2016 20:15:51 by David Roberts (211.26.51.197)