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Section 4.5 Fermat Numbers

Definition 4.5.1.

For \(n \in\mathbb{N}_0\) we call the number \(F_n = 2^{2^{n}}+1\) the \(n\)th Fermat numbers. Those which are prime are called Fermat primes.

This conjecture is FALSE. In 1732, Euler showed that

\begin{equation*} F_5 = 2^{2^{5}}+1=4294967297 = 641\cdot 6700417. \end{equation*}

A lot of computation has been done regarding this conjecture, though there is not yet much hope for a definitive answer. There is a heuristic argument for the finiteness of Fermat primes.

Using the Prime Number Theorem we know that the probability that a number \(x\) is prime is about \(C/\log(x),\) where \(C\) is some constant. Thus the expected number of Fermat primes is at most

\begin{equation*} C\sum_{n=0}^{\infty}\frac{1}{\log(F_n)}=C\sum_{n=0}^{\infty}\frac{1}{\log(2^{2^{n}}+1)}\lt C\sum_{n=0}^{\infty}\frac{1}{\log(2^{2^n})}=\frac{C}{\log(2)}\sum_{n=0}^{\infty}2^{-n}=\frac{2C}{\log(2)}. \end{equation*}

If is also interesting to note the importance to geometry. In 1801, Gauss showed that a regular polygon with \(k\) sides can be constructed a with straight edge and compass if and only if \(k = 2^ep_1...p_r\) where \(p_1,..., p_r\) are Fermat primes.