haupt-exponent | modulo index | modulo order | residue index

Let n be a positive number having primitive roots. If g is a primitive root of n, then the numbers 1, g, g^2, ..., g^(ϕ(n) - 1) form a reduced residue system modulo n, where ϕ(n) is the totient function. In this set, there are ϕ(ϕ(n)) primitive roots, and these are the numbers g^c, where c is relatively prime to ϕ(n). The smallest exponent e for which b^e congruent 1 (mod n), where b and n are given numbers, is called the multiplicative order (or sometimes haupt-exponent or modulo order) of b (mod n). The multiplicative order is implemented in the Wolfram Language as MultiplicativeOrder[g, n].

Carmichael function | complete residue system | congruence | discrete logarithm | full reptend prime | out-shuffle | polynomial order | primitive root | suborder function