I don't speak English well, but I have a serious problem.

Let $\displaystyle k > 1$. The sequence is defined by recursive equation $\displaystyle a_0 = 1,

a_{n+1} = 10^{k}a_n+1$ for $\displaystyle n>=0$.

We must prove that this sequence contains only finitely many primes.

Thank you for our answers.