Permutable prime

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

In base 10, all the permutable primes with less than 49081 digits are (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, 1111111111111111111, 11111111111111111111111, R₃₁₇, R₁₀₃₁

where R_n = (10ⁿ-1)/9 is the number with n ones.

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; 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.

References

1. ^ a b H. E. Richert, "On permutable primtall," Norsk Matematiske Tiddskrift 33 (1951), 50–54.

2. ^ T. Bhargava & P. Doyle, "On the existence of absolute primes," Math. Mag. 47 (1974), 233.

3. ^ Chris Caldwell, The Prime Glossary: permutable prime at The Prime Pages.

4. ^ A.W. Johnson, "Absolute primes," Mathematics Magazine 50 (1977), 100-103.