How can I know if $2^{2^{2^{2^{2}}}}+1=?$ is prime?

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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