**Background:**

Fermat numbers are numbers of the form

*F*(

*n*) = 2^(2^

*n*)+1, where

*n*is a whole number.

**Fermat numbers can be generated using a number of recurrence relations, each of which can be proven using mathematical induction. When Fermat first studied them in the 17th Century, he noticed many properties that distinguish them from other groups of numbers. Including the following:**

- Every prime of the form 2^
*n*+1 is a Fermat number, and such primes are called Fermat primes. This is true, provided that*n*is a power of 2. - The recurrence relation
*F*(*n*) =*F*(0)**F*(1)**F*(2)****F*(*n*-1)+2 can help deduce Goldbach's Theorem (no two Fermat numbers share a common factors). - No Fermat number can be expressed as the sum of two primes, with the exception of
*F*(1) = 2+3. - With the exception of
*F*(0) and*F*(1), every Fermat number ends with a 7. - The sum of the reciprocals of all the Fermat numbers is irrational.

Your Task:

Your Task:

Using the above information, the

*Read More*sources provided, and possibly a math software (Mathematica, Maple, etc.), investigate these properties of Fermat numbers. In doing so, try to answer these questions:

- Is
*F*(*n*) composite for all*n*>4? - Are there infinitely many Fermat primes?
- Are there infinitely many composite Fermat numbers?