gives the multiplicative order of k modulo n, defined as the smallest integer such that .
gives the generalized multiplicative order of k modulo n, defined as the smallest integer such that for some .
- MultiplicativeOrder is also known as modulo order or haupt‐exponent.
- Integer mathematical function, suitable for both symbolic and numerical manipulation.
- Typically used in modular arithmetic and cryptography.
- MultiplicativeOrder[k,n] gives the smallest positive integer m such that the remainder when dividing km by n is equal to 1.
- MultiplicativeOrder returns unevaluated if there is no integer satisfying the necessary conditions.
- For a FiniteFieldElement object a, MultiplicativeOrder[a] gives the multiplicative order of a, defined as the smallest positive integer m such that is the multiplicative identity of the finite field.
Examplesopen allclose all
Basic Examples (2)
Basic Applications (5)
Number Theory (4)
Properties & Relations (5)
Possible Issues (1)
For nonzero integers k and n, MultiplicativeOrder[k,n] exists if and only if k and n are coprime:
Interactive Examples (1)
MultiplicativeOrder of each integer below a given prime number:
Neat Examples (2)
Ulam spiral of MultiplicativeOrder:
Wolfram Research (1999), MultiplicativeOrder, Wolfram Language function, https://reference.wolfram.com/language/ref/MultiplicativeOrder.html (updated 2023).
Wolfram Language. 1999. "MultiplicativeOrder." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/MultiplicativeOrder.html.
Wolfram Language. (1999). MultiplicativeOrder. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MultiplicativeOrder.html