Is this a Prime Number?

There was an interesting conversation I found on the CodeVision Discord server

\[2^{2^{77,232,917}} - 1\]

Now we have this number, but it’s obviously huge, so we don’t want to calculate all those digits out to determine if it is prime or not. However, it is worth noting that for some $n\in\mathbb{N}$,

\[\forall n>2,\;4\mid2^{n}\]

Note the following sequence

\[\begin{aligned} 2^4 &= 16 \\ 2^8 &= 256 \\ 2^{12} &= 4096\\ &... \end{aligned}\]

Since $2$ raised to any multiple of $4$ (besides $0$, obviously) results in an integer with a $6$ as the one’s digit. We can then say

\[\forall n>2,\;5\mid2^{2^n} - 1\]

Therefore, $2^{2^{77,232,917}} - 1$ is not prime.

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