TITLE: For primes sufficiently large, must digit products be zero?
QUESTION [12 upvotes]: Let $\{P_n\}, n\in \mathbb{N}$ be the sequence prime numbers such that $P_1=2, P_2=3\dots$.
Define a new sequence $\{M_n\}$,  $n\in \mathbb{N}$, such that $M_n=$Product of the digits of the $nth$ prime in its decimal representation. 
Now the question is: Does there exist an $N\in  \mathbb{N}$, such that $M_n=0$ $  \forall n\geq N$ ?

REPLY [11 votes]: No.  This 2016 paper by James Maynard  says that there are infinitely many primes not containing the digit $0$.  So $M_n \gt 0$ infinitely often.