Permutations and primes

Abstract

The problem of N-digit sets all permutations of which give primes is discussed. Such sets may include only digits 1, 3, 7 and 9, and none of 0, 2, 5, 4, 6, 8. Direct calculations show that such full-permutation digit sets occur at N = 1, 2, 3 and are absent in the 4 - 10 interval of N. On the other hand the formal full permutation at N = 19, 23, 317 and 1031 (as well at N = 2) cases is provided by repunits (integers all digits of which are 1). The existence/nonexistence of other (not repunits) full-permutation digit sets for arbitrary large N is an open question with probable negative answer. The maximal-permutation digit sets with maximal number of primes are given for N = 4 - 10.