For many variants of this question the answer seems to be not known but at least this question in the comments
More generally, are there infinitely many primes such that at least one non-trivial permutation of their digits preserves the primality? – Sylvain JULIEN
in binary has the answer Yes. Thanks to @AlexeiKulikov for the improved argument.
Use
- (i) Prime number theorem
- (i ii) Pigeonhole principle
By (i) there are at least $c (2^n/n)$ primes in $[2^{n-1},2^n)$ whose number of 1s is in the interval $[1,n]$.
By (ii) there are two primes in $[2^{n-1},2^n)$ with the same number of 1s.