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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 analytic number theory and related branches of mathematics, a complex-valued arithmetic function: is a Dirichlet character of modulus (where is a positive integer) if for all integers and : [1]
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 ...
In field theory, a primitive element of a finite field GF(q) is a generator of the multiplicative group of the field. In other words, α ∈ GF( q ) is called a primitive element if it is a primitive ( q − 1) th root of unity in GF( q ) ; this means that each non-zero element of GF( q ) can be written as α i for some natural number i .
In number theory, Artin's conjecture on primitive roots states that a given integer a that is neither a square number nor −1 is a primitive root modulo infinitely many primes p. The conjecture also ascribes an asymptotic density to these primes. This conjectural density equals Artin's constant or a rational multiple thereof.
In general, if m has prime power factors , then one may find an root of unity mod m by finding primitive roots of unity mod for each prime and pulling them all back via the Chinese remainder theorem. This gives an n t h {\displaystyle n^{th}} root of unity ω {\displaystyle \omega } such that ω n / 2 = − 1 mod m {\displaystyle \omega ^{n/2 ...
Mazur's torsion theorem (algebraic geometry) Mean value theorem ; Measurable Riemann mapping theorem (conformal mapping) Mellin inversion theorem (complex analysis) Menelaus's theorem ; Menger's theorem (graph theory) Mercer's theorem (functional analysis) Mermin–Wagner theorem ; Mertens's theorems (number theory)
The term axiomatic geometry can be applied to any geometry that is developed from an axiom system, but is often used to mean Euclidean geometry studied from this point of view. The completeness and independence of general axiomatic systems are important mathematical considerations, but there are also issues to do with the teaching of geometry ...