Twice prime number?

A twice prime number is defined as a prime number whose digits are also prime. For example: $23$ is prime. It is made up of the digit $2$, which is prime, and the digit $3$, which is also prime. Therefore, $23$ is twice prime. Counter example: $19$ is prime, but $1$ nor $9$ is prime and therefore $19$ is not twice prime.

Does twice prime only consist of two prime digits? Or can we have a three digit prime which can qualify as twice prime?

• $113,137,173$ are some of them – user35508 Jun 25 '17 at 13:35
• Next questions: are there infinitely many twice primes? And also, does this concept already exist under a different guise? Finally, I feel uncomfortable that this is a concept rooted in our decimal number system. – Colm Bhandal Jun 25 '17 at 13:37
• @user35508 $1$ is not considered a prime number- I believe that's what is alluded to above. – Colm Bhandal Jun 25 '17 at 13:38
• @ColmBhandal ... It is also made up like $11$ and $3$ – user35508 Jun 25 '17 at 13:39
• @user35508 Very good point. But my interpretation of the OPs question is that "digit" means single decimal digit. We must await clarification from the OP...? – Colm Bhandal Jun 25 '17 at 13:40

$$233, 257, 2377, 23327,\ldots.$$