Prime-bounded subwords

Abstract

In the number $373$ all subwords ($3$, $7$, $37$, $73$, and $373$) are prime. Similarly, in $9719$ all subwords are divisible by at most one prime. And similarly again in $7319797913$ all subwords are divisible by at most two primes. These are the largest integers with their respective properties. We show for any $k\ge 1$ there are only finitely many integers having subwords divisible by at most $k$ primes. In fact, we show for any $B$ and $d$ coprime that $n$ contains a base-$B$ subword divisible by $d$ if $n>B^d$. So as example consequence, past a certain point every prime contains a subword divisible by say $10000000007$.