# Does the string of prime numbers contain all natural numbers?

Does the string of prime numbers contain every natural number as its sub-string?

If $d$ is the number we want to find, define $s=10d+1$. By definition, $\gcd(s,10)=1$ and $s$ contains the digits of $d$.
Dirichlet’s Theorem’s implies there’s a prime of the form $p:=s+k \times 10^n$ where $10^n$ is chosen so that it has as many zeroes as digits of $s$. The digits of $d$ appear in the digits of $p$, and thus in the given string of primes.