Fermat prime

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If $n$ is a nonnegative integer the $n^{th}$ Fermat number is defined to be $F_n = 2^{2^n}+1$.

If $F_n$ is prime, then it is known as a Fermat prime. The first $5$ Fermat numbers (for $n=0,1,2,3,4$) are known to be prime. Indeed: \begin{align*} F_0 &= 2^{2^0}+1 = 3\\ F_1 &= 2^{2^1}+1 = 5\\ F_2 &= 2^{2^2}+1 = 17\\ F_3 &= 2^{2^3}+1 = 257\\ F_4 &= 2^{2^4}+1 = 65537. \end{align*} Based on these results, one might conjecture (as did Fermat) that all Fermat numbers are prime. However, this fails for $n=5$: $F_5 = 2^{2^5}+1 = 4294967297 = 641 \cdot 6700417$. In fact the primes listed above are the only Fermat numbers known to be prime.

Fermat primes are also the only primes in the form $2^m+1$. This is easy to see, if $m$ has an odd factor $a>1$ then $2^{m/a}+1|2^m+1$, a contradiction, hence $m$ is a power of $2$.

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