Permutable prime

From Wikipedia, the free encyclopedia

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, the all permutable primes with less than 4 digits are (with the permutations listed in parentheses):

2, 3, 5, 7, 11, 13(31), 17(71), 37(73), 79(97), 113(131, 311), 199(919, 991), 337(373, 733)

It is obvious that all permutable primes of two or more digits are composed from the digits 1, 3, 7, 9. It is proved^[3] 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.

Any repunit prime can automatically be assumed to be a permutable prime as well. 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. The generalization can safely be made that for any positional number system, permutable primes can only have digits that are coprime with the number system's radix.

[edit] References

^ H. E. Richert, "On permutable primtall," Norsk Matematiske Tiddskrift 33 (1951), 50--54.
^ T. Bhargava & P. Doyle, "On the existence of absolute primes," Math. Mag. 47 (1974), 233.
^ A.W. Johnson, "Absolute primes," Mathematics Magazine 50 (1977), 100-103.