Referring to Is it true, that every prime (except 2) can be found as a divisor of enough long series of 1-s? , I have the same question.
I have the intuitive hyptohesis, that every prime can be found as a divisor of at least one of 11, 101, 1001, 10001, 100001, ...
Is it true? My try to prove that didn't work in the related question.
The prime $31$ is not a divisor of any number of the form $10\cdots01$.
Indeed, a prime $p$ divides $10^n+1$ iff the multiplicative order of $10 \bmod p$ is even (and equal to $2n$). But this does not always happen. For $p=31$, we have $10^{15} \equiv 1 \bmod 31$ and so the multiplicative order of $10 \bmod 31$ is not even, since it has to divide $15$. (The order is actually $15$.)
