When is the number $11111\cdots1$ a prime number? For which $n$ is the sum:
$$\sum_{k=0}^{n}10^k$$
a prime number? Are they finite?
 A: Partial solution:
Given a prime $p \neq 2, 5$, let $n = Q(p - 1) + R$. We use Fermat's Little Theorem modulo $p$:
$$\begin{align}
\sum_{k = 0}^n10^k &= 1 + \sum_{i = 1}^Q10^{(i - 1)(p - 1)}(10^1 + 10^2 + \dotsb + 10^{p - 1}) + 10^{Q(p - 1)}(10^1 + 10^2 + \dotsb + 10^R)\\
&\equiv 1 + \sum_{i = 1}^Q\frac{p(p - 1)}{2} + (10^1 + 10^2 + \dotsb + 10^R)\\
&\equiv \sum_{k = 0}^R10^k \pmod{p}.
\end{align}$$
The $p(p - 1)/2$ came from the fact that the residues of $10^1, 10^2, \dotsc, 10^{p - 1}$ modulo $p$ form a permutation of $1, 2, \dotsc, (p - 1)$.
A: After adding $1$, the $n$s for which the sum above is known to be prime or probably prime are given in the OEIS sequence A004023.  (The $+1$ is because the OEIS lists numbers $n$ such that $(10^n-1)/9$ is prime, but the sum in the question is instead equal to $(10^{n+1}-1)/9$.)  Also, see the Wikipedia article and the Prime Pages entry.
These primes are called "repunit primes" since their decimal expansion consists of a repeated series of $1$s.   The repunits corresponding to
$$
n=1, 18, 22, 316, \text{and } 1030 
$$
$$\text{ (using the $n$ in the questioner's formula above)}
$$
are known to be prime.  The repunits corresponding to
$$
n=49080, 86452, 109296, \text{and } 270342
$$
$$\text{ (again using the $n$ in the questioner's formula above)}
$$
are as far as I know only known to be probably prime.
The obvious factorization
$$
\frac{10^{km}-1}{9}=\frac{10^k-1}{9}(1+10^k+\cdots+10^{k(m-1)})
$$
means that, for $(10^{n+1}-1)/9$ to be prime, $n+1$ must also be prime.
There are conjectured to be infinitely many repunit primes.
A: Numbers having the shape n = 1111......1 have less probability to be a prime.
If sum of all digits are divisible by 3 then that number is divisible by 3. Using this rule If number of digits of n is divisible by 3 then n is surely not a prime. $\therefore 111,111111,111111111, ....$ are divisible by 3
If number of digits are divisible by 2 (even number of digits in n), then n can be partitions into multiple 11s. e.g. 1111 can be partitioned as 11 11. Which is divisible by 11 $\therefore 11,1111,111111, .... $ are divisible by 11.
So if number of digits are not divisible by 2 or 3 (e.g. $5,7,11,13,17,19,23,25,29,31,35,37,....$) we are not obvious that the number is not prime. However for these numbers we get large prime divisors most of the time. Like 11111 = 41*271
