Prime-bounded subwords

TL;DR

This paper investigates integers with subwords that are prime or divisible by a limited number of primes, establishing finiteness results and bounds for such integers, and demonstrating that large numbers necessarily contain subwords divisible by large primes.

Contribution

It proves finiteness of integers with subwords divisible by at most k primes and provides bounds for subword divisibility in large numbers.

Findings

Only finitely many integers have subwords divisible by at most k primes.

For any base B and coprime d, integers larger than B^d contain a subword divisible by d.

Large integers necessarily contain subwords divisible by large primes.

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 $10000000007$.