How can I prove analytically the number $2^{100000}+1$ is not prime??
How can I prove analytically the number
$$(2^{100000}+1)$$ is not prime??
Let $x=2^{20000}$. Then $$2^{100000}+1=x^5+1=(x+1)(x^4-x^3+x^2-x+1).$$
We prove a more general statement: a number of the form $a^n+1$ where $n$ is not a power of two is not prime.
Proof: let $n=2^mk$ with $k$ odd. now let $j=a^{2^m}$. Then $a^n+1=j^k+1$.
Since $k$ is odd we have $j^k+1=(j+1)(j^{k-1}-j^{k-2}+j^{k-3}-\dots+j-1)$