### Mathematics Research Projects

## 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.

Previous |
Home |
Glossary |
Next

**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).