Fermat prime

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A Fermat prime is a prime number of the form $2^{2^n} + 1$ for some nonnegative integer $n$.

A number of the form $2^{2^n} + 1$ for nonnegative integer $n$ is a Fermat number. The first five Fermat numbers (for $n=0,1,2,3,4$) are \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*} and each of these is a Fermat prime. 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.

Primes one more than a power of 2

Fermat primes are also the only primes in the form $2^m+1$.

Proof

Suppose that $m$ has an odd factor $a > 1$. For all odd $a$, we have by the Root-Factor Theorem that $x + 1$ divides $x^a + 1$. Since this is true as a statement about polynomials, it is true for every integer value of $x$. In particular, setting $x = 2^{m / a}$ gives that $2^{m / a} + 1 | 2^m + 1$, and since $2^m + 1 > 2^{m / a} + 1 > 1$, this shows that $2^m + 1$ is not prime.

It follows that if $2^m + 1$ is prime, $m$ must have no odd factors other than $1$ and so must be a power of 2.

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