Primitive root information

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...

In mathematics, a primitive root may mean: Primitive root modulo n in modular arithmetic Primitive nth root of unity amongst the solutions of zn = 1 in...

2\not \equiv 4{\pmod {4}}.} Let z be a primitive nth root of unity. A power w = zk of z is a primitive ath root of unity for a = n gcd ( k , n ) , {\displaystyle...

divisors modulo n. A primitive root modulo n, is a generator of the group of units of the ring of integers modulo n. There exist primitive roots modulo n if...

permutation group Primitive root of unity; See Root of unity Primitive triangle, an integer triangle whose sides have no common prime factor Primitive (phylogenetics)...

In mathematics, the term primitive element can mean: Primitive root modulo n, in number theory Primitive element (field theory), an element that generates...

such that a0 ≠ 0 (mod p), has at most m roots. Primitive root modulo m: A number g is a primitive root modulo m if, for every integer a coprime to m,...

{\displaystyle (\mathbb {Z} /n\mathbb {Z} )^{\times }} is called a primitive root modulo n. If there is any generator, then there are φ ( φ ( n ) ) {\displaystyle...

choice of primitive root, by Wilhelm Patz. Jacobi's original tables use 10 or –10 or a number with a small power of this form as the primitive root whenever...

contains a nth primitive root of unity if and only if n is a divisor of q − 1; if n is a divisor of q − 1, then the number of primitive nth roots of unity...

(OEIS: A088165) Primes p for which the least positive primitive root is not a primitive root of p2. Three such primes are known; it is not known whether...

n ≥ p. If b is a primitive root of p, and p does not divide x or |x - y|, then n is a multiple of p - 1. (Since b is a primitive root mod p and p does...

In field theory, the primitive element theorem states that every finite separable field extension is simple, i.e. generated by a single element. This...

Euler's totient function. ζ n {\displaystyle \zeta _{n}} is a complex primitive n-th root of unity: ζ n n = 1 , {\displaystyle \zeta _{n}^{n}=1,} but ζ n ≠...

"the index of a to the base r modulo m") for r x ≡ a (mod m) if r is a primitive root of m and gcd(a,m) = 1. Discrete logarithms are quickly computable in...

on the fact that e − 2 π i / n {\textstyle e^{-2\pi i/n}} is an n'th primitive root of unity, and thus can be applied to analogous transforms over any finite...

only if 10 is a primitive root modulo n. In particular, it follows that L(p) = p − 1 if and only if p is a prime and 10 is a primitive root modulo p. Then...

one can find all the primitive root quadruples. The following Python code demonstrates this algorithm, producing the primitive root quadruples listed above...

multiplier a is an element of high multiplicative order modulo m (e.g., a primitive root modulo n), and the seed X0 is coprime to m. Other names are multiplicative...

implying that n is prime. Conversely, if n is prime, then there exists a primitive root modulo n, or generator of the group (Z/nZ)*. Such a generator has order...

the first uncountable ordinal number (also sometimes written as Ω) A primitive root of unity, like the complex cube roots of 1 The Wright Omega function...

fields), it is sufficient to choose α {\displaystyle \alpha } as a primitive nth root of unity, which replaces the condition (1) by: α k ≠ 1 {\displaystyle...

rational numbers of any primitive nth-root of unity ( e 2 i π / n {\displaystyle e^{2i\pi /n}} is an example of such a root). An important relation linking...