Intereting Posts

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I could calculate the following prime numbers

$$2+1=3$$

$$2^{2}+1=5$$

$$2^{2^{2}}+1=17$$

$$2^{2^{2^{2}}}+1=65537$$

Are the following numbers prime??? $$2^{2^{2^{2^{2}}}}+1=?$$ $$2^{2^{2^{2^{2^{2}}}}}+1=??$$

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Note that

$$

2^{2^{2^{2^2}}}+1=2^{2^{16}}+1=F_{16},

$$

the $16$-th Fermat number, and

$$

2^{2^{2^{2^{2^2}}}}+1=2^{2^{65536}}+1=F_{65536}.

$$

All Fermat numbers $F_n$ for $5\le n \le 32$ are known to be composite (Wikipedia). Summaries of factoring status are given here. Specifically, $F_{16}$ is known to be divisible by

$$

1575\cdot 2^{19} + 1 = 825753601

$$

and

$$

180227048850079840107 \cdot 2^{20} + 1 = 188981757975021318420037633.

$$

As far as I can tell, nothing is known about the primality of $F_{65536}$.

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