# Permutable Prime with more than 3 Digits is Probably Repunit

## Conjecture

Let $p$ be a permutable prime with more than $3$ digits.

Then $p$ is very probably repunit.

Hence the next permutable prime after $991$ is the repunit prime $R_{19}$:

$1 \, 111 \, 111 \, 111 \, 111 \, 111 \, 111$

## Historical Note

It is trivially true, as shown in Digits of Permutable Prime, that a permutable prime with more than $1$ digit can contain only the digits from the set $\set {1, 3, 7, 9}$.

T.N. Bhargava and P.H. Doyle demonstrated in a $1974$ paper that a permutable prime cannot contain all of $1$, $3$, $7$ and $9$.

Subsequently, Allan W. Johnson showed that a non-repunit permutable prime is of the form $\sqbrk {aaa \cdots aab}$ and have more than $9$ billion digits.

In $1995$, Dmitry Mavlo determined further restrictions on the structure of permutable primes.

However, it still has not been proven that a permutable prime with more than $3$ digits is always a repunit prime.