  James Maynard has taken the analytic number theory world by storm in the last
decade, proving several important and surprising theorems, resolving questions
that had seemed far out of reach. He is perhaps best known for his work on
small and large gaps between primes (which were discussed, hot off the press,
in my 2015 CEB lecture). In this article we will discuss two other Maynard
breakthroughs: -- Mersenne numbers take the form $2^n-1$ and so appear as
$111\dots 111$ in base 2, having no digit `$0$'. It is a famous conjecture that
there are infinitely many such primes. More generally it was, until Maynard's
work, an open question as to whether there are infinitely many primes that miss
any given digit, in any given base. We will discuss Maynard's beautiful ideas
that went into partly resolving this question. -- In 1926, Khinchin gave
remarkable conditions for when real numbers can usually be ``well
approximated'' by infinitely many rationals. However Khinchin's theorem
regarded 1/2, 2/4, 3/6 as distinct rationals and so could not be easily
modified to cope, say, with approximations by fractions with prime
denominators. In 1941 Duffin and Schaefer proposed an appropriate but
significantly more general analogy involving approximation only by reduced
fractions (which is much more useful). We will discuss its recent resolution by
Maynard together with Dimitris Koukoulopoulos.
