In number theory, given a positive integer n and an integer a coprime to n, the multiplicative order of a modulo n is the smallest positive integer k such that .
In other words, the multiplicative order of a modulo n is the order of a in the multiplicative group of the units in the ring of the integers modulo n.
The order of a modulo n is sometimes written as ord<sub>n</sub>(a).
Example
The powers of 4 modulo 7 are as follows:
: <math>\begin{array}{llll}
4^0 &= 1 &=0 \times 7 + 1 &\equiv 1\pmod7 \\
4^1 &= 4 &=0 \times 7 + 4 &\equiv 4\pmod7 \\
4^2 &= 16 &=2 \times 7 + 2 &\equiv 2\pmod7 \\
4^3 &= 64 &=9 \times 7 + 1 &\equiv 1\pmod7 \\
4^4 &= 256 &=36 \times 7 + 4 &\equiv 4\pmod7 \\
4^5 &= 1024 &=146 \times 7 + 2 &\equiv 2\pmod7 \\
\vdots\end{array}</math>
The smallest positive integer k such that is 3, so the order of is 3.
Properties
Even without knowledge that we are working in the multiplicative group of integers modulo n, we can show that a actually has an order by noting that the powers of a can only take a finite number of different values modulo n, so according to the pigeonhole principle there must be two powers, say s and t and without loss of generality s > t, such that a<sup>s</sup> ≡ a<sup>t</sup> (mod n). Since a and n are coprime, a has an inverse element a<sup>−1</sup> and we can multiply both sides of the congruence with a<sup>−t</sup>, yielding .
The concept of multiplicative order is a special case of the order of group elements. The multiplicative order of a number a modulo n is the order of a in the multiplicative group whose elements are the residues modulo n of the numbers coprime to n, and whose group operation is multiplication modulo n. This is the group of units of the ring Z<sub>n</sub>; it has φ(n) elements, φ being Euler's totient function, and is denoted as U(n) or U(Z<sub>n</sub>).
As a consequence of Lagrange's theorem, the order of always divides φ(n). If the order of a is actually equal to φ(n), and therefore as large as possible, then a is called a primitive root modulo n. This means that the group U(n) is cyclic and the residue class of a generates it.
The order of a (mod n) also divides λ(n), a value of the Carmichael function, which is an even stronger statement than the divisibility of φ(n).
Programming languages
- Maxima CAS: zn_order (a, n)
- Wolfram Language: MultiplicativeOrder[k, n]
- Rosetta Code – examples of multiplicative order in various languages
See also
- Discrete logarithm
- Modular arithmetic