Smallest 3-Digit Permutable Prime

Theorem
The smallest $3$-digit permutable prime is $113$.

Proof

 * $113$ is prime.


 * $131$ is prime.


 * $311$ is prime.

Consider the $3$-digit primes smaller than $113$:
 * $101, 103, 107, 109$

They all contain a zero.

Thus, for each of these, at least one permutation ends in a zero.

Hence it is divisible by $10$ and so is not prime.