# 3-Digit Permutable Primes

## Theorem

The $3$-digit permutable primes are:

$311, 199, 337$

and their anagrams, and no other.

## Proof

It is confirmed that:

$113, 131, 311$ are all prime
$199, 919, 991$ are all prime
$337, 373, 733$ are all prime.

From Digits of Permutable Prime, all permutable primes contain digits in the set:

$\left\{ {1, 3, 7, 9}\right\}$

The sum of a $3$-digit repdigit number is divisible by $3$.

By Divisibility by 3 it follows that all $3$-digit repdigit numbers are divisible by $3$ and therefore composite.

Hence the possibly permutable primes are:

$113, 117, 119, 133, 137, 139, 177, 179, 199, 337, 339, 377, 379, 399, 779, 799$

and their anagrams.

We eliminate $113, 199, 337$ from this list, as it has been established that they are permutable primes.

Of those remaining, the following are composite:

 $\displaystyle 117$ $=$ $\displaystyle 3^2 \times 13$ $\displaystyle 119$ $=$ $\displaystyle 7 \times 17$ $\displaystyle 133$ $=$ $\displaystyle 7 \times 19$ $\displaystyle 177$ $=$ $\displaystyle 3 \times 59$ $\displaystyle 339$ $=$ $\displaystyle 3 \times 113$ $\displaystyle 377$ $=$ $\displaystyle 13 \times 29$ $\displaystyle 779$ $=$ $\displaystyle 19 \times 41$ $\displaystyle 799$ $=$ $\displaystyle 17 \times 47$

It remains to demonstrate that at least one anagram of the remaining numbers:

$137, 139, 179, 379$

is composite.

We find that:

 $\displaystyle 371$ $=$ $\displaystyle 7 \times 53$ $\displaystyle 319$ $=$ $\displaystyle 11 \times 29$ $\displaystyle 791$ $=$ $\displaystyle 7 \times 113$ $\displaystyle 793$ $=$ $\displaystyle 13 \times 61$

All contenders are eliminated except for the established permutable primes $113, 119, 337$ and their anagrams.

$\blacksquare$