2-Digit Permutable Primes

Theorem

The $2$-digit permutable primes are:

$11, 13, 17, 37, 79$

and their anagrams, and no other.

Proof

It is confirmed that:

$13$ and $31$ are both prime

$17$ and $71$ are both prime

$37$ and $73$ are both prime.

$79$ and $97$ are both prime.

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

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

By Prime Repdigit Number is Repunit, all $2$-digit repdigit numbers are composite except for $11$ which is prime.

From Repunit Prime is Permutable Prime, it follows that $11$ is a permutable prime.

Apart from $11$ whose status is known, the possibly permutable primes are:

$13, 17, 19, 37, 39, 79$

and their anagrams.

We eliminate $13, 17, 37, 79$ from this list, as it has been established that they are permutable primes.

Of those remaining, the following is composite:

\(\ds 39\)

\(=\)

\(\ds 3 \times 13\)

It remains to demonstrate that the anagram of $19$, that is, $91$, composite.

We find that:

\(\ds 91\)

\(=\)

\(\ds 7 \times 13\)

All contenders are eliminated except for the established permutable primes $11, 13, 17, 37, 79$ 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): $113$