A number of the form 2

^{2n} + 1, for any

natural number n, sometimes denoted as F

_{n}. The first five (

3,

5,

17,

257 and

65537) are primes, and some people accuse

Pierre de Fermat of claiming that all such numbers were prime.

Of course, the general consensus used to be that this was true, until 1732, when Leonhard Euler discovered that 2^32 + 1 can be expressed as 641 * 6700417. In 1880, a retired mathematician named Landry discovered that 2^64 + 1 = 274177 * 67280421310721. In 1970, John Brillhart (my math professor) and his associate Morrison discovered that 2^128+ 1 = 59649589127497217 * 5704689200685129054721. After that, the prime factors start to get incredibly large. It is now highly suspected that all Fermat numbers greater than F_{4} are composite; the smallest Fermat number whose primality is unknown is presently 2^(2^31) + 1. The largest prime factor for a Fermat number was recently discovered by John Cosgrave and Yves Gallot on July 23, 1999: They found that the prime number 3*2^382449+1 divides the Fermat number 2^(2^382447) + 1.

Detailed information on the Cosgraves-Gallot discovery can be found at:

*http://perso.wanadoo.fr/yves.gallot/papers/nlkff.html*

The present status of the Fermat numbers can be attained by directing your internet browsing software to:

*http://www.prothsearch.net/fermat.html*