TITLE: Proving that $(1+10^n)$cannot be a prime number when $(n>2)$
QUESTION [7 upvotes]: Proving that $$(1+10^n)$$ cannot  be a prime number 
when $n>2$

REPLY [13 votes]: The number $$10^n+1$$ can only be prime if $n$ is of the form $2^k$.
A simple proof of this fact
Suppose, $n$ has an odd prime factor $p$. Denote $q:=\frac{n}{p}$
Then $10^q+1$ is a non-trivial factor of $10^n+1=10^{qp}+1=(10^q)^p+1$
because for every number $t$ dividing $10^q+1$, we get $10^n+1=(10^q)^p+1\equiv (-1)^p+1\equiv 0\ (\ mod\ t\ )$
The generalized fermat numbers have been studied deeply. The smallest
number $10^n+1$, for which it is NOT known if it is composite or prime,
is 
$$10^{2^{24}}+1$$
The numbers $10^{2^k}+1$ with $2\le k\le 23$ are composite.
So, a prime of the form $10^n+1$ with $n>2$ would have at least a magnitude comparable
to the largest known prime. It would have at least $2^{24}+1=16,777,217$ digits!
See here :
http://www.prothsearch.net/GFN10.html
for more informations