multiplicative order of an integer modulo m


Let m>1 be an integer and let a be another integer relatively prime to m. The order ( of a modulo m (or the multiplicative orderMathworldPlanetmath of amodm) is the smallest positive integer n such that an1modm. The order is sometimes denoted by orda or ordma.


Several remarks are in order:

  1. 1.

    Notice that if gcd(a,m)=1 then a belong to the units (/m)× of /m. The units (/m)× form a group with respect to multiplicationPlanetmathPlanetmath, and the number of elements in the subgroupMathworldPlanetmathPlanetmath generated by a (and its powers) is the order of the integer a modulo m.

  2. 2.

    By Euler’s theorem, aϕ(m)1modm, therefore the order of a is less or equal to ϕ(m) (here ϕ is the Euler phi function).

  3. 3.

    The order of a modulo m is precisely ϕ(m) if and only if a is a primitive rootMathworldPlanetmath for the integer m.

Title multiplicative order of an integer modulo m
Canonical name MultiplicativeOrderOfAnIntegerModuloM
Date of creation 2013-03-22 16:20:38
Last modified on 2013-03-22 16:20:38
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 5
Author alozano (2414)
Entry type Definition
Classification msc 13-00
Classification msc 13M05
Classification msc 11-00
Synonym multiplicative order
Related topic PrimitiveRoot
Related topic PrimeResidueClass