TITLE: Does the string of prime numbers contain all natural numbers?
QUESTION [67 upvotes]: Does the string of prime numbers $$2357111317\ldots$$ contain every natural number as its sub-string?

REPLY [4 votes]: Let $n$ be a natural number, and let $\mathcal N(n)$ be the set of natural numbers not containing $n$ as a substring when written in base 10. It is well-known that $\sum_{k\in \mathcal N(n)}\frac{1}{k}$ converges.
Let $\mathcal P$ be the set of prime numbers. It is also well-known that $\sum_{p \in\mathcal P}\frac{1}{p}$ diverges.
Therefore it cannot be the case that $ \mathcal P\subseteq\mathcal N(n)$, and $n$ is therefore a substring of some prime (of infinitely many primes, in fact).