multiplicative order of an integer modulo m
Definition.
Let be an integer and let be another integer relatively prime to . The order (http://planetmath.org/OrderGroup) of modulo (or the multiplicative order of ) is the smallest positive integer such that . The order is sometimes denoted by or .
Remarks.
Several remarks are in order:
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1.
Notice that if then belong to the units of . The units form a group with respect to multiplication, and the number of elements in the subgroup generated by (and its powers) is the order of the integer modulo .
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2.
By Euler’s theorem, , therefore the order of is less or equal to (here is the Euler phi function).
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3.
The order of modulo is precisely if and only if is a primitive root for the integer .
Title | multiplicative order of an integer modulo m |
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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 |