It is well-known that not only does the arithmetic progression $\{a+kd\}_{k \in \mathbb{Z}^{+}}$ contain infinitely many prime numbers, but also that the series of the reciprocals of those primes diverges. The answer to the OP's question can be obtained from the following general result:
If $\{a_{i}\}_{i \in \mathbb{N}}$ is a strictly increasing sequence of natural numbers such that the series $\sum_{i=1}^{\infty} \frac{1}{a_{i}}$ diverges, then the decimal fraction
$$\alpha = 0.(a_{1})(a_{2})\ldots (a_{n}) \ldots$$
is irrational.