# 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.

$\blacksquare$

## Sources

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