A positive integer is prime if it is not 1 and all its only integer
factors are 1 and itself. Given a positive integer with at least
two digits one can cyclically rotate its digits or permute its
digits. For example, given the number 134 the numbers which arise
from it by cyclic rotation are: 341 and 413. The numbers which
arise from it by permutation are: 134, 143, 341, 314, 413, 431.
A number is called cyclically prime if all of its cyclic
rotations are prime and it is called permutation prime
if all its permutations are prime.
Problems:
1. Are there cyclic primes with d digits for every d?
2. Are there permutation primes with d digits for every d?
3. Are there infinitely many cyclic primes?
4. Are there infinitely many permutation primes?
Extensions:
1. A number is called cyclically divisible by d if all
its cyclic rotations are divisible by d. For a given positive
integer d are there numbers cyclically divisible by d? (Similar
definition: permutably divisible by d.)
2. For a given positive integer n, c(n) (p(n)) the number of cyclic
rotations (permutations) of the digits of n which are prime.
Which n have c(n) = 0 (p(n) = 0)? Investigate the functions c(n)
and p(n).
3. Study similar questions where the cyclic rotations or permutations
of n are examined with regard to being divisible by a particular
integer d.
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