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.

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $113$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $113$