I was trying to solve a question on number theory. It says, given a range (start, end $ \le 10^{15}$ and end - start $ \le 10^9$), how many prime digit prime numbers exist? Generating prime digits numbers is easy but to check whether this number is prime, it runs out of time. I discarded numbers ending with $2$ or $5$, still a large number to go through primality test. So is there an efficient way to do primality test for prime digits prime number.
Note: I tried with sieve but failed for large number typically $\ge10^7$.