## 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.

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).