TITLE: 3-7 primes in base 10
QUESTION [12 upvotes]: After a quick look at the sequence (of primes) A020463,
$$
3, 7, 37, 73, 337, 373, 733, 773, 3373, \dots,
$$
the following question is straighforward:
Are there infinitely many primes compiled from digits 3 and 7 in base 10?
As any question about primes, this should be tough. Nevertheless I believe there exists some reasonable heuristics to answer it.

REPLY [11 votes]: Significant progress on this question has been made by Maynard. In this preprint on the arXiv he proves that: 

"There are infinitely many primes that do not contain the digit $9$ in their decimal expansion."

This represents the current state of knowledge on this problem, and combines various methods and techniques. From the first page of Maynard's paper:

"Our proof is based on a combination of the circle method, Harman’s sieve, the method of bilinear sums, the large sieve, the geometry of numbers and a comparison with a Markov process. In particular, we make key use of the Fourier structure of [the set of integers without that do not have a $9$ in their decimal expansion]."

See also this earlier paper by Maynard on primes not containing a certain digit modulo $q$.