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A '''permutable prime''' is a [[prime number]] which, in a given base, can have its digits' positions switched through any [[permutation]] and still be a prime number. [[H. E. Richert]], who is supposed to be the first to study these primes, called them permutable primes,<ref name="Richert">H. E. Richert, "On permutable primtall," ''Norsk Matematiske Tiddskrift'' '''33''' (1951), 50–54.</ref> but later they were also called '''absolute primes'''.<ref>T. Bhargava & P. Doyle, "On the existence of absolute primes," ''Math. Mag.'' '''47''' (1974), 233.</ref>

In [[base 10]], all the permutable primes with fewer than 49,081 digits are known {{OEIS|id=A003459}}:
:[[2 (number)|2]], [[3 (number)|3]], [[5 (number)|5]], [[7 (number)|7]], [[11 (number)|11]], [[13 (number)|13]], [[17 (number)|17]], [[31 (number)|31]], [[37 (number)|37]], [[71 (number)|71]], [[73 (number)|73]], [[79 (number)|79]], [[97 (number)|97]], [[113 (number)|113]], [[131 (number)|131]], [[199 (number)|199]], 311, 337, 373, 733, 919, 991, R<sub>19</sub> (1111111111111111111), R<sub>23</sub>, R<sub>317</sub>, R<sub>1031</sub>.

Of the above, there are 16 unique permutation sets, with smallest elements
:2, 3, 5, 7, R<sub>2</sub>, 13, 17, 37, 79, 113, 199, 337, R<sub>19</sub>, R<sub>23</sub>, R<sub>317</sub>, R<sub>1031</sub>.

Note R<sub>''n''</sub> = <math>\tfrac{10^n-1}{9}</math> is a [[repunit]], a number consisting only of ''n'' ones (in [[base 10]]). Any [[repunit prime]] is a permutable prime with the above definition, but some definitions require at least two distinct digits.<ref>Chris Caldwell, [http://primes.utm.edu/glossary/page.php?sort=PermutablePrime The Prime Glossary: permutable prime] at The [[Prime Pages]].</ref>

All permutable primes of two or more digits are composed from the digits 1, 3, 7, 9, because no prime number except 2 is even, and no prime number besides 5 is divisible by 5.  It is proved<ref>A.W. Johnson, "Absolute primes," ''Mathematics Magazine'' '''50''' (1977), 100–103.</ref> that no permutable prime exists which contains three different of the four digits 1, 3, 7, 9, as well as that there exists no permutable prime composed of two or more of each of two digits selected from 1, 3, 7, 9.

There is no ''n''-digit permutable prime for 3 < ''n'' < 6·10<sup>175</sup> which is not a repunit.<ref name="Richert"/> It is [[conjecture]]d that there are no non-repunit permutable primes other than those listed above.

In base 2, only repunits can be permutable primes, because any 0 permuted to the one's place results in an even number. Therefore the base 2 permutable primes are the [[Mersenne prime]]s. The generalization can safely be made that for any [[positional number system]], permutable primes with more than one digit can only have digits that are [[coprime]] with the [[radix]] of the number system. One-digit primes, meaning any prime below the radix, are always trivially permutable.

== References ==
<references />

{{DEFAULTSORT:Permutable Prime}}
[[Category:Base-dependent integer sequences]]
[[Category:Classes of prime numbers]]
[[Category:Permutations]]