  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.
