Project 22: Permutation Primes and Cyclic Primes

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.

Joseph Malkevitch
Department of Mathematics and Computing
York College (CUNY)
Jamaica, New York 11451-0001
email: malkevitch@york.cuny.edu
(Comments and results related to the project above are welcome.)

Acknowledgements
Some of this work was prepared with partial support from the National Science Foundation (Grant Number: DUE 9555401) to the Long Island Consortium for Interconnected Learning (administered by SUNY at Stony Brook, Alan Tucker, Project Director).