From Dirichlet's theorem on arithmetic progressions, if $\text{gcd}(a,b)=1$ we know $\{ak+b\}_{k\ge 0}$ contains infinitely many primes. Let those primes be $p_1,p_2,\cdots$. Then the real $$\alpha=0.p_1p_2p_3\cdots\tag{1}$$ is it irrational? Here the primes are placed side by side, as in $p_1=13,p_2=53,\cdots$ then the expression would be like $\alpha=0.1353\cdots$. So to rephrase my question, is $\alpha$ irrational, as defined in $(1)$? Thanks for answering.