Haralambos Terzidis and George Danas

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 8, 2002, Number 1, Pages 1–20

**Full paper (PDF, 7791 Kb)**

## Details

### Authors and affiliations

Haralambos Terzidis

*Educational Institution of Thessaloniki School of Sciences – Department of Mathematics
P.O.Box 14561, GR-54101 Thessaloniki, Greece
*

George Danas

*Educational Institution of Thessaloniki School of Sciences – Department of Mathematics*

P.O.Box 14561, GR-54101 Thessaloniki, Greece

P.O.Box 14561, GR-54101 Thessaloniki, Greece

### Abstract

We detail and study the periods of the inverses of integers in order to understand the behavior of periodic arithmetic sequences whose terms are the digits of the inverses of integers. More precisely, by introducing the rank of a prime number we show how to compute the period of the inverse of prime numbers of variable rank. Furthermore, we prove some properties and a number of statements concerning the residue class 10 (mod *n*), (10, *n*) = 1, which lead to define the degree of a prime number. In addition, by using both the rank and the degree of a prime number we compute the periods of the inverses of powers of prime numbers. Finally, we study the behavior of the simple recurring decimals and we establish our main result which shows how to compute the period and its length of the inverse of an integer.

### Keywords

- Primitive root
- Residue class
- Reduced residue system
- Rank and degree of a prime number

### 2010 Mathematics Subject Classification

- 11A07
- 11A15

### References

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- L.-K. Hua, Introduction to Number Theory, Springer-Verlag, New York, 1982.
- A. Hurwitz, N. Kritikos, Lectures on Number Theory, Springer-Verlag, New York, 1986.
- J. H. McClellan, CM. Rader, Number Theory in Digital Signal Processing, Prentice-Hall, Englewood Cliffs NJ, 1979.
- M. R. Schroeder, Number Theory in Science and Communication, 2nd ed., Springer-Verlag, New York, 1990.
- H. Terzidis, G. Danas, The spectral analysis of the periods of the inverses of integers, J. Inst. Math. Comput. Sci. (Comp. Sci. Ser.) 10 1 (1999), 61-69.

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## Cite this paper

Terzidis, H., & Danas, G. (2002). The periods of the inverses of integers. *Notes on Number Theory and Discrete Mathematics*, 8(1), 1-20.