<s>[INST] Decomposition of $2+2^2+2^{2^2}+2^{2^{2^2}}+2^{2^{2^{2^2}}}$ It's a little question about number theory :
Remark that we have :
$$2+2^2=2*3$$
$$2+2^2+2^{2^2}=2*11$$
$$2+2^2+2^{2^2}+2^{2^{2^2}}=2*32779$$
Do we have :
$$2+2^2+2^{2^2}+2^{2^{2^2}}+2^{2^{2^{2^2}}}=2*p\quad?$$
Where $p$ is a prime number .
Thanks a lot for your comment or answer .
 [/INST] No.$$2+2^2+2^{2^2}+2^{2^{2^2}}+2^{2^{2^{2^2}}} \equiv 2 + 1 + 1 +1 + 1 \equiv 0\pmod{3}$$ Because $2^{2n}\equiv (-1)^{2n}\equiv 1\pmod{3}$
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