# Digits of primes

# Digits of primes

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Many famous open questions about primes - often of a recreational flavour - can be interpreted as questions about the digits of primes in a given base. For example, Mersenne primes are primes with no digit equal to zero in base 2, and it is completely open whether there are infinitely many such primes. We will talk about recent work showing there are infinitely many primes with no digit equal to 7 in their decimal expansion. (And similarly with 7 replaced by any other digit.) This shows the existence of primes in a 'thin' set of numbers (sets which contain at most X^{1-c} elements less than X) which is typically very difficult.

The proof relies on a fun mixture of tools including Fourier analysis, Markov chains, Diophantine approximation, combinatorial geometry as well as tools from analytic number theory.