Multiplicative order

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 ${\textstyle a^{k}\ \equiv \ 1{\pmod {n}}}$ .^[1]

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 $\operatorname {ord} _{n}(a)$ .^[2]

Example

The powers of 4 modulo 7 are as follows:

{\begin{array}{llll}4^{0}&=1&=0\times 7+1&\equiv 1{\pmod {7}}\\4^{1}&=4&=0\times 7+4&\equiv 4{\pmod {7}}\\4^{2}&=16&=2\times 7+2&\equiv 2{\pmod {7}}\\4^{3}&=64&=9\times 7+1&\equiv 1{\pmod {7}}\\4^{4}&=256&=36\times 7+4&\equiv 4{\pmod {7}}\\4^{5}&=1024&=146\times 7+2&\equiv 2{\pmod {7}}\\\vdots \end{array}}

The smallest positive integer k such that 4^k ≡ 1 (mod 7) is 3, so the order of 4 (mod 7) 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^s ≡ a^t (mod n). Since a and n are coprime, a has an inverse element a⁻¹ and we can multiply both sides of the congruence with a^−t, yielding a^s−t ≡ 1 (mod n).

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_n; it has φ(n) elements, φ being Euler's totient function, and is denoted as U(n) or U(Z_n).

As a consequence of Lagrange's theorem, the order of a (mod n) 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)^[3]
Wolfram Language : MultiplicativeOrder[k, n]^[4]
Rosetta Code - examples of multiplicative order in various languages^[5]

References

^ Niven, Zuckerman & Montgomery 1991, Section 2.8 Definition 2.6
^ von zur Gathen, Joachim; Gerhard, Jürgen (2013). Modern Computer Algebra (3rd ed.). Cambridge University Press. Section 18.1. ISBN 9781107039032.
^ Maxima 5.42.0 Manual: zn_order
^ Wolfram Language documentation
^ rosettacode.org - examples of multiplicative order in various languages

Niven, Ivan; Zuckerman, Herbert S.; Montgomery, Hugh L. (1991). An Introduction to the Theory of Numbers (5th ed.). John Wiley & Sons. ISBN 0-471-62546-9.