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Permutable prime

A permutable prime, also known as anagrammatic 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, but later they were also called absolute primes.
In base 10, all the permutable primes with fewer than 49,081 digits are known
:2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, R19 (1111111111111111111), R23, R317, R1031, ...
Of the above, there are 16 unique permutation sets, with smallest elements
:2, 3, 5, 7, R2, 13, 17, 37, 79, 113, 199, 337, R19, R23, R317, R1031, ...
Note R''n'' = $\tfrac$ 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.〔Chris Caldwell, (The Prime Glossary: permutable prime ) at The Prime Pages.〕
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〔A.W. Johnson, "Absolute primes," ''Mathematics Magazine'' 50 (1977), 100–103.〕 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'' < 6·10175 which is not a repunit.〔 It is conjectured 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 ones place results in an even number. Therefore the base 2 permutable primes are the Mersenne primes. 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.
In duodecimal, the permutable primes are (unique permutation sets, with smallest elements)
:2, 3, 5, 7, Ɛ, R2, 15, 57, 5Ɛ, R3, 117, 11Ɛ, 555Ɛ, R5, R17, R81, R91, R225, R255, R4ᘔ5, ...
In base 10, every permutable prime is a repunit or a near-repdigit, that is, it is a permutation of the integer
''P''(''b'', ''n'', ''x'', ''y'') = ''xxxx''...''xxxy''''b'' (''n'' digits, in base ''b'')
where ''x'' and ''y'' are digits which is coprime to ''b'', if ''x'' = ''y'', then ''x'' = ''y'' = 1.
This is not true in all bases, but exceptions are rare and could be finite in any given base; the only exceptions below 109 in bases up to 20 are: 13911, 36A11, 24713, 78A13, 29E19 (M. Fiorentini, 2015).
Let ''P''(''b'', ''n'', ''x'', ''y'') be a permutable prime in base ''b'' and let ''p'' be a prime such that ''n'' > ''p''. If ''b'' is a primitive root of ''p'', and ''p'' does not divide ''x'', then ''n'' is a multiple of ''p'' - 1.
== References ==

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