International Journal of Mathematics and Mathematical Sciences
Volume 26 (2001), Issue 10, Pages 589-596

Arithmetic progressions that consist only of reduced residues

Paul A. Tanner III

Department of Mathematics, University of South Florida, 4202 E. Fowler Avenue, Tampa 33620, FL, USA

Received 13 June 2000; Revised 25 February 2001

Copyright © 2001 Paul A. Tanner III. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper contains an elementary derivation of formulas for multiplicative functions of m which exactly yield the following numbers: the number of distinct arithmetic progressions of w reduced residues modulo m; the number of the same with first term n; the number of the same with mean n; the number of the same with common difference n. With m and odd w fixed, the values of the first two of the last three functions are fixed and equal for all n relatively prime to m; other similar relations exist among these three functions.