Fermat numbers. Are they all prime?
It was once believed that all numbers of the form $$2^{2^n}+1$$ were prime until Euler disproved this by showing that $$2^{2^5}+1$$ is not prime. Is there a formula or a way to know which of the Fermat numbers are prime?.
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$\begingroup$The only known Fermat primes so far are the cases $n = 0,1,2,3,4$. You can get some more information here.
$\endgroup$ 3 $\begingroup$Something like a formula exists in Pepin's Test, whereby $F_n=2^{2^n}+1$ is prime iff $3^{2^{n-1}}\equiv -1\bmod F_n$. In principle, this is simple to implement as it involves just repeated squaring. But for Fermat numbers of current interest the numbers are so large that the computation becomes nontrivial.
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