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Complete Residue System

A set of numbers $a_0$, $a_1$, ..., $a_{m-1}$ (mod $m$) form a complete set of residues, also called a covering system, if they satisfy

a_i\equiv i\ \left({{\rm mod\ } {m}}\right)

for $i=0$, 1, ..., $m-1$. For example, a complete system of residues is formed by a base $b$ and a modulus $m$ if the residues $r_i$ in $b^i\equiv r_i\ \left({{\rm mod\ } {m}}\right)$ for $i=1$, ..., $m-1$ run through the values 1, 2, ..., $m-1$.

See also Exact Covering System, Haupt-Exponent


Guy, R. K. ``Covering Systems of Congruences.'' §F13 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 251-253, 1994.

© 1996-9 Eric W. Weisstein