3-Digit Permutable Primes

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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