On the proximity of large primes

TL;DR

This paper demonstrates that there are infinitely many pairs of primes close to each other under various digit-based metrics, including differences in base $q$ expansions and digit similarity in initial segments.

Contribution

It introduces new results on the distribution of primes with respect to digit-based proximity in various numeral systems, using a sphere-packing argument.

Findings

Infinitely many prime pairs differ in at most two digits in base $q$.

Existence of infinitely many primes sharing the first $t$ digits.

For infinitely many integers, many primes differ by at most one digit in base $q$.

Abstract

By a sphere-packing argument, we show that there are infinitely many pairs of primes that are close to each other for some metrics on the integers. In particular, for any numeration basis $q$, we show that there are infinitely many pairs of primes the base $q$ expansion of which differ in at most two digits. Likewise, for any fixed integer $t,$ there are infinitely many pairs of primes, the first $t$ digits of which are the same. In another direction, we show that, there is a constant $c$ depending on $q$ such that for infinitely many integers $m$ there are at least $c\log \log m$ primes which differ from $m$ by at most one base $q$ digit.