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A permutable prime is a prime number, which, in a given base, can have its digits switched to any possible permutation and still spell a prime number. H. E. Richert, who supposedly first studied these primes, called them permutable primes, but later they were also called absolute primes._{n} = $tfrac\{10^n-1\}\{9\}$ is the number with n ones.## References

In base 10, all the permutable primes with less than 49081 digits are :

- 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, 1111111111111111111, 11111111111111111111111, R
_{317}, R_{1031}

Any repunit prime is a permutable prime with the above definition, but some definitions require at least two distinct digits.

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 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^{175} 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 one's place results in an even number; unless we consider 1 a prime number and 10 permutable with 01. 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 permutable.

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Last updated on Saturday October 11, 2008 at 16:29:55 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Saturday October 11, 2008 at 16:29:55 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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