### Mathematics Research Projects

## Project 16: Repeating Decimals

If one computes 1/n where n is a positive integer sometimes one
obtains a pattern of digits where there is no initial string of
digits but only a repeating pattern. (For example: 1/3 or 1/7).
If the period of the repeating pattern is n-1 one can show that
n must be prime. (For example, 17 and 19.) However, there are
many primes p for which the repeat period is less than p-1. (For
example 11 or 13.)

**Problems**

1. For each integer i is there a prime which has repeat period
i?

2. Is there any relation between the digits in the repeated pattern
and the prime or the digits in the prime?

3. Is there a pattern in the values of the primes for which the
repeats have the same length k?

**Extensions**

1. Examine questions such as the above where the number is expressed
in a base other than 10.

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).