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If = is an odd number (/) is cyclic of order (); a generator is called a primitive root mod . [14] Let g q {\displaystyle g_{q}} be a primitive root and for ( a , q ) = 1 {\displaystyle (a,q)=1} define the function ν q ( a ) {\displaystyle \nu _{q}(a)} (the index of a {\displaystyle a} ) by
In number theory, a kth root of unity modulo n for positive integers k, n ≥ 2, is a root of unity in the ring of integers modulo n; that is, a solution x to the equation (or congruence) (). If k is the smallest such exponent for x, then x is called a primitive kth root of unity modulo n. [1]
In modular arithmetic, a number g is a primitive root modulo n if every number a coprime to n is congruent to a power of g modulo n. That is, g is a primitive root modulo n if for every integer a coprime to n, there is some integer k for which g k ≡ a (mod n). Such a value k is called the index or discrete logarithm of a to the base g modulo n.
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory , the theory of group characters , and the discrete Fourier transform .
Weisstein, Eric W. "Primitive Root". MathWorld. Web-based tool to interactively compute group tables by John Jones; OEIS sequence A033948 (Numbers that have a primitive root (the multiplicative group modulo n is cyclic)) Numbers n such that the multiplicative group modulo n is the direct product of k cyclic groups:
Prime-counting function: Number of primes less than or equal to a given number. Partition function: Order-independent count of ways to write a given positive integer as a sum of positive integers. Möbius μ function: Sum of the nth primitive roots of unity, it depends on the prime factorization of n. Prime omega functions; Chebyshev functions
In mathematics, a primitive root may mean: Primitive root modulo n in modular arithmetic; Primitive nth root of unity amongst the solutions of z n = 1 in a field; See ...
Both 2 and 3 are primitive λ-roots modulo 5 and also primitive roots modulo 5. n = 8. The set of numbers less than and coprime to 8 is {1,3,5,7} . Hence φ(8) = 4 and λ(8) must be a divisor of 4. In fact λ(8) = 2 since ().