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,[1] but later they were also called absolute primes.[2]

In base 10, all the permutable primes with fewer than 49,081 digits are known (sequence A003459 in OEIS):

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 Rn = $\tfrac{10^n-1}{9}$ 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.[3]

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[4] 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.[1] 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 one's 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.

References

1. ^ a b Richert, Hans-Egon (1951). "On permutable primtall". Norsk Matematiske Tiddskrift 33: 50–54. Zbl 0054.02305.
2. ^ Bhargava, T.N.; Doyle, P.H. (1974). "On the existence of absolute primes". Math. Mag. 47: 233. Zbl 0293.10006.
3. ^ Chris Caldwell, The Prime Glossary: permutable prime at The Prime Pages.
4. ^ A.W. Johnson, "Absolute primes," Mathematics Magazine 50 (1977), 100–103.