In 1877 Pepin proved the following theorem for deciding
if Fermat numbers are prime (this is one of the nicest
examples of the classical primality proving tests):
Jacobi symbol (k|Fn)
is -1. These include k=3, 5, and 10.
If Fn is prime, this primality can be shown by Pepin's test, but when Fn is composite, Pepin's test does not tell us what the factors will be (only that it is composite). For example, Selfridge and Hurwitz showed that F14 was composite in 1963, but it was not until 2010 that its first factor was found.
Related pages (outside of this work)