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Topics referred to by the same term
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
Primitive_root
Number with an integer power equal to 1
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
Root_of_unity
Modular arithmetic concept
number g is a primitive root modulo n if every number a coprime to n is congruent to a power of g modulo n. In symbols, g is a primitive root modulo n if
Primitive_root_modulo_n
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
Root_of_unity_modulo_n
Generator of the multiplicative group of a finite field
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)
Primitive element (finite field)
Primitive_element_(finite_field)
Conjecture 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
Artin's conjecture on primitive roots
Artin's_conjecture_on_primitive_roots
Computation modulo a fixed integer
(mod p) has at most d non-congruent solutions. Primitive root modulo m: A number g is a primitive root modulo m if, for every integer a coprime to m,
Modular_arithmetic
Spreading of sound energy
in either one or two directions. Primitive-root diffusors are based on a number theoretic sequence based on primitive roots. Although they produce a notch
Diffusion_(acoustics)
Class of prime numbers
multiplicative order ordp b = p − 1, which is equivalent to b being a primitive root modulo p. The term "long prime" was used by John Conway and Richard
Full_reptend_prime
Complex-valued arithmetic function
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 ≠
Dirichlet_character
(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
List_of_prime_numbers
Group of units of the ring of integers modulo n
{\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
Multiplicative group of integers modulo n
Multiplicative_group_of_integers_modulo_n
Method of exchanging cryptographic keys
multiplicative group of integers modulo p, where p is prime, and g is a primitive root modulo p. To guard against potential vulnerabilities, it is recommended
Diffie–Hellman_key_exchange
Topics referred to by the same term
permutation group Primitive root of unity; See Root of unity Primitive triangle, an integer triangle whose sides have no common prime factor Primitive (phylogenetics)
Primitive
Irreducible polynomial whose roots are nth roots of unity
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
Cyclotomic_polynomial
Discrete Fourier transform algorithm
on the fact that e − 2 π i / n {\textstyle e^{-2\pi i/n}} is an nth primitive root of unity, and thus can be applied to analogous transforms over any finite
Fast_Fourier_transform
Topics referred to by the same term
In mathematics, the term primitive element can mean: Primitive root modulo n, in number theory Primitive element (field theory), an element that generates
Primitive_element
Algebraic structure
every n p {\displaystyle np} th root of unity is also a n {\displaystyle n} th root of unity. It follows that primitive n p {\displaystyle np} th roots
Finite_field
{\displaystyle p} be an odd prime, and let g {\displaystyle g} be a primitive root modulo p {\displaystyle p} . Let x 0 {\displaystyle x_{0}} be a seed
Blum–Micali_algorithm
Prime pair of the form (p, 2p+1)
except −1 (if nonresidue), is a primitive root. It follows that for a safe prime, the least positive primitive root is a prime number. With the exception
Safe and Sophie Germain primes
Safe_and_Sophie_Germain_primes
Type of linear congruential generator with no additive constant
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
Lehmer random number generator
Lehmer_random_number_generator
Field theory theorem
In field theory, the primitive element theorem states that every finite separable field extension is simple, i.e. generated by a single element. This
Primitive_element_theorem
Fractal composed of tangent circles
one can find all the primitive root quadruples. The following Python code demonstrates this algorithm, producing the primitive root quadruples listed above
Apollonian_gasket
Natural number
is the only odd prime p {\displaystyle p} known for which 2 is not a primitive root of 4 p 2 + 1 {\displaystyle 4p^{2}+1} . It is the thirteenth Pierpont
193_(number)
Natural number
On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Wolfram MathWorld; Primitive Root Wikimedia Commons has media related to 191 (number). v t e
191_(number)
Last letter of the Greek alphabet
including 0 (sometimes written ω 0 {\displaystyle \omega _{0}} ) A primitive root of unity, like the complex cube roots of 1 The Wright Omega function
Omega
Discrete Fourier transform for prime sizes
groups is that there exists a generator of the group (sometimes called a primitive root, which can be found by exhaustive search or slightly better algorithms)
Rader's_FFT_algorithm
1839 mathematical tables by Carl Jacobi
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
Canon_arithmeticus
Minimal polynomial of a primitive element in a finite field
of degree m with coefficients in GF(p) = Z/pZ is a primitive polynomial if it is monic and has a root α in GF(pm) such that { 0 , 1 , α , α 2 , α 3 , …
Primitive polynomial (field theory)
Primitive_polynomial_(field_theory)
Decimal representation of a number whose digits are periodic
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
Repeating_decimal
Relationship between the rational roots of a polynomial and its extreme coefficients
product of primitive polynomials. Now any rational root p/q corresponds to a factor of degree 1 in Q[X] of the polynomial, and its primitive representative
Rational_root_theorem
Algorithm for checking if a number is prime
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
Lucas_primality_test
Natural number
N. J. A. (ed.). "Sequence A001913 (Full reptend primes: primes with primitive root 10)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation
229_(number)
Function in mathematical number theory
whose order equals the exponent, λ(n). Such an element is called a primitive λ-root modulo n. The Carmichael function is named after the American mathematician
Carmichael_function
Music recording studio in Berry Hill, Tennessee, US
Diffusor Systems founder Peter D'Antonio, Ph.D., Studio C features a primitive root sequence diffusor made up of 138,646 individual pieces of wood. Studio
Blackbird_Studio
Circle-like pointset in a geometric plane
t14 + t18 + t22 + t26) + η20(t8 + t20) + η6(t12 + t24), where η is a primitive root of GF(32) satisfying η5 = η2 + 1. As the hyperovals in the Desarguesian
Oval_(projective_plane)
Type of algebraic field extension
\mathbb {Q} ({\sqrt[{3}]{2}}).} Let ω {\displaystyle \omega } be a primitive cubic root of unity. Then since, Q ( 2 3 ) = { a + b 2 3 + c 4 3 ∈ Q ¯ | a
Normal_extension
Generalisation of Fourier transform to any ring
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
Discrete Fourier transform over a ring
Discrete_Fourier_transform_over_a_ring
Concept in modular arithmetic
equal to φ(n), and therefore as large as possible, then a is called a primitive root modulo n. This means that the group U(n) is cyclic and the residue class
Multiplicative_order
{\displaystyle K} is finite, then so is L {\displaystyle L} , and any primitive root of L {\displaystyle L} will generate the field extension. If K {\displaystyle
Steinitz's theorem (field theory)
Steinitz's_theorem_(field_theory)
Error-correcting codes
make the code cyclic. In particular, if α {\displaystyle \alpha } is a primitive root of the field F {\displaystyle F} , then by definition all non-zero elements
Reed–Solomon_error_correction
in 1973 that αb,c is b-normal whenever c is an odd prime and b is a primitive root of c2. In 2002, Bailey & Crandall showed that coprimality of b, c >
Stoneham_number
Pair of polynomial sequences
i {\displaystyle x-g_{i}} where each g i {\displaystyle g_{i}} is a primitive root of unity. Thus, we obtain: x n C n ( x + 1 x ) = ∏ d ≥ 3 , d ∣ 4 n
Chebyshev_polynomials
Ties Legendre symbols to permutation signatures
numbers mod p, which is a cyclic group of order p − 1. The jth power of a primitive root modulo p will have index the greatest common divisor i = (j, p − 1)
Zolotarev's_lemma
Branch of the Afroasiatic languages
in some cases counting). The primitive root ṣ-f and the trilateral root stems m-ṣ-f, ṣ-h-f, and ṣ-f-r are used. This root also exists in other Semitic
Semitic_languages
Equation for radii of tangent circles
reduction is possible. A root quadruple is said to be primitive if it has no nontrivial common divisor. Every primitive root quadruple can be found from
Descartes's_theorem
Reflex actions in infants
Primitive reflexes are reflex actions originating in the central nervous system that are exhibited by normal infants, but not neurologically intact adults
Primitive_reflexes
Equations of degree 5 or higher cannot be solved by radicals
{\displaystyle K_{i}} that extends F i − 1 {\displaystyle F_{i-1}} by a primitive root of unity, and one redefines F i {\displaystyle F_{i}} as K i ( x i )
Abel–Ruffini_theorem
Field extension of the rational numbers by a primitive root of unity
{\displaystyle \zeta _{n}=e^{2\pi i/n}\in \mathbb {C} .} This is a primitive n {\displaystyle n} th root of unity. Then the n {\displaystyle n} th cyclotomic field
Cyclotomic_field
Fungus-plant symbiotic association
consensus among paleomycologists that mycorrhizal fungi served as a primitive root system for early terrestrial plants. This is because, prior to plant
Mycorrhiza
Natural number
N. J. A. (ed.). "Sequence A001913 (Full reptend primes: primes with primitive root 10.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation
181_(number)
Theorem on prime numbers
for which the product is −1 are precisely the ones where there is a primitive root modulo m. Wilson prime Table of congruences Agoh–Giuga conjecture Because
Wilson's_theorem
Polynomial in which all coefficients are one
and 2 is a primitive root modulo m + 1 (over GF(p) with prime p, it is irreducible if and only if m + 1 is prime and p is a primitive root modulo m +
All_one_polynomial
Mathematical group that can be generated as the set of powers of a single element
group under multiplication. It is cyclic, since it is generated by the primitive root z = 1 2 + 3 2 i = e 2 π i / 6 : {\displaystyle z={\tfrac {1}{2}}+{\tfrac
Cyclic_group
Basal organ of a vascular plant
vigorous root systems are essential for crop stability and prevention of lodging. Absorption of water and mineral nutrients. Root epidermal cells and root hairs
Root
Algorithm for determining whether a number is prime
number a modulo n is n − 1 for a prime n when a is a primitive root modulo n. If we can show a is primitive for n, we can show n is prime. Riesel (1994) pp
Primality_test
American actress
Chem. Soc. 2013, 135(16):6092-9. Erdos, P. and Shapiro H.N., On The Least Primitive Root Of A Prime, 1957, euclidproject.org. Eunice Cho at IMDb v t e
Eunice_Cho
totient function Noncototient Nontotient Euler's theorem Wilson's theorem Primitive root modulo n Multiplicative order Discrete logarithm Quadratic residue Euler's
List_of_number_theory_topics
Mathematical conjecture about zeros of L-functions
guaranteed to run in polynomial time. For every prime p there exists a primitive root mod p (a generator of the multiplicative group of integers modulo p)
Generalized Riemann hypothesis
Generalized_Riemann_hypothesis
Describes statistically the splitting of primes in a given Galois extension of Q
extensions, obtained from the field of rational numbers by adjoining a primitive root of unity of a given order. For example, the ordinary integer primes
Chebotarev_density_theorem
Positive integer of the form (2^(2^n))+1
multiply this by a number A, which is greater than the square root of P and is a primitive root modulo P (i.e., it is not a quadratic residue). Then take
Fermat_number
Function in discrete mathematics
N = e − i 2 π / N {\displaystyle \omega _{N}=e^{-i2\pi /N}} is a primitive Nth root of unity. For example, in the case when N = 2 {\displaystyle N=2}
Discrete_Fourier_transform
Power of a prime number
numbers. Every prime power excluding powers of 2 greater than 4 has a primitive root; thus the multiplicative group of integers modulo pn (that is, the group
Prime_power
Integer whose multiples are digit rotations
specifically, this sequence is the set of primes p such that b is a primitive root modulo p. A conjecture of Emil Artin is that this sequence contains
Cyclic_number
Cuban-American mathematician
Brauer, with a thesis on Elementary estimates for the least positive primitive root modulo pr. After finishing her master's degree, she became a secondary
Miriam_Leiva
Integer side lengths of a right triangle
(3, 4, 5) is a primitive Pythagorean triple whereas (6, 8, 10) is not. Every Pythagorean triple can be scaled to a unique primitive Pythagorean triple
Pythagorean_triple
Mathematical group based upon a finite number of elements
of this group is as the complex nth roots of unity. Sending a to a primitive root of unity gives an isomorphism between the two. This can be done with
Finite_group
Theoretical object in mathematics
the cyclic group of order n, the isomorphism depending on choice of a primitive root of unity: F 1 n = μ n . {\displaystyle \mathbf {F} _{1^{n}}=\mu _{n}
Field_with_one_element
Concept that is not defined in terms of previously defined concepts
In mathematics, logic, philosophy, and formal systems, a primitive notion is a concept that is not defined in terms of previously defined concepts. It
Primitive_notion
Theorems that help decompose a finite group based on prime factors of its order
m}&0\\0&x^{jm}\end{bmatrix}}} , x is any primitive root of Fq. Since the order of Fq is q − 1, its primitive roots have order q − 1, which implies that
Sylow_theorems
Filtration of the Galois group of a local field extension
where ζ {\displaystyle \zeta } is a p n {\displaystyle p^{n}} -th primitive root of unity, can be described explicitly: G s = Gal ( K n / K e ) , {\displaystyle
Ramification_group
Mathematical tree of integer right triangles
tree of primitive Pythagorean triples is a mathematical tree in which each node represents a primitive Pythagorean triple and each primitive Pythagorean
Tree of primitive Pythagorean triples
Tree_of_primitive_Pythagorean_triples
Points with distinct displacement vectors
by Lloyd R. Welch. The Welch–Costas array is constructed by taking a primitive root g of a prime number p and defining the array A by A i , j = 1 {\displaystyle
Costas_array
{\displaystyle p^{n}} , generated by any choice of a primitive pnth root of unity ζpn. Since all of the primitive roots in μ p n {\displaystyle \mu _{p^{n}}} are
Cyclotomic_character
Lists of values of mathematical functions
by employing Newton's method in the complex plane to solve for the primitive root of zN − 1). This method would produce an exact table in exact arithmetic
Trigonometric_table
Umbrella term for deadly disease, especially of livestock
word in Hebrew is דֶּבֶר "dever" (Strong's #01698), derived from the primitive root "dabar" in the sense of "to destroy." In some parts of Scotland, force-fire
Murrain
Field extension generated by a one element
is a root of an irreducible polynomial of degree n in K [ X ] {\displaystyle K[X]} . However, in the case of finite fields, the term primitive element
Simple_extension
Conditions in number theory
division is to let g be a primitive root (mod p); then the first set is all the numbers whose indices with respect to this root are ≡ 0 (mod 4), the second
Quartic_reciprocity
Austrian mathematician (1898–1962)
group; and the second the frequency with which a given integer a is a primitive root modulo primes p, when a is fixed and p varies. These are unproven; in
Emil_Artin
Concept in quantum information theory
factor ω {\displaystyle \omega } . If ω {\displaystyle \omega } is a primitive root of unity, for example ω ≡ e 2 π i d {\displaystyle \omega \equiv e^{\frac
Mutually_unbiased_bases
Fast Fourier transform algorithm
{\displaystyle N-1} and ω N {\displaystyle \omega _{N}} denotes the primitive root of unity: ω N = e − 2 π i N , {\displaystyle \omega _{N}=e^{-{\frac
Split-radix_FFT_algorithm
Operation on formal power series
the a t h {\displaystyle a^{th}} primitive root of unity. Then we have the following formula, often known as the root of unity filter: ∑ n ≥ 0 f a n +
Generating function transformation
Generating_function_transformation
Conditions under which the congruence x^3 equals p (mod q) is solvable
to let e be a primitive root (mod p); then the first (resp. second, third) set is the numbers whose indices with respect to this root are congruent to
Cubic_reciprocity
Formal power series
generally, suppose that a ≥ 3 and that ωa = exp 2πi/a denotes the ath primitive root of unity. Then, as an application of the discrete Fourier transform
Generating_function
Unique positive real number which when multiplied by itself gives 2
The square root of 2 (approximately 1.4142) is the positive real number that, when multiplied by itself or squared, equals the number 2. It may be written
Square_root_of_2
Natural number
N. J. A. (ed.). "Sequence A001913 (Full reptend primes: primes with primitive root 10.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation
300_(number)
Ancient Greek goddess
royal appellation Artemas of Xenophon". Charles Anthon argued that the primitive root of the name is probably of Persian origin from *arta, *art, *arte, all
Artemis
Type of prime number
{\displaystyle \pm 1} term is positive if and only if n {\displaystyle n} has a primitive root and negative otherwise. For every natural number n {\displaystyle n}
Wilson_prime
American journalist (born 1967)
an unusual-looking uncircumcised penis that Dreher described as a "primitive root wiener". Dreher said he intended to continue blogging and might also
Rod_Dreher
Fourier-related mathematical transform
algorithm is the constraint that each dimension of the transform has a primitive root. Hartley, Ralph V. L. (March 1942). "A More Symmetrical Fourier Analysis
Discrete_Hartley_transform
^{ij},0\leq j\leq N-1,} where α {\displaystyle \alpha } is the N-th primitive root of 1 in G F ( p m ) {\displaystyle \mathrm {GF} (p^{m})} . If the polynomial
Cyclotomic fast Fourier transform
Cyclotomic_fast_Fourier_transform
Rational numbers with root 5 added
subfield of Q ( ζ ) {\displaystyle \mathbb {Q} (\zeta )} . For any primitive root of unity ζ n {\displaystyle \zeta _{n}} , the maximal real subfield
Golden_field
Village in Zgharta District, Lebanon
village. Strong's Hebrew/Greek Dictionary, entry 7682, 'sagab/saw-gab': a primitive root; to be (causatively, make) lofty, especially inaccessible; by implication
Kfarsghab
Russian mathematician (1937–2008)
{\displaystyle n\leq x} , for which ( n + a ) {\displaystyle (n+a)} is a primitive root modulo q {\displaystyle q} , one gets an asymptotic expression of the
Anatoly_Karatsuba
Construction on any polygon that yields a regular polygon with the same number of sides
= ( 1 − ωσj )−1( S − ωσj I ) Aj , where ω = exp( 2πi/n ) is the nth primitive root of unity and σj is the jth term of a permutation σ of the integer sequence
Petr–Douglas–Neumann_theorem
Function whose domain is the positive integers
prime)}}.\end{cases}}} See Multiplicative group of integers modulo n and Primitive root modulo n. 2 ω ( n ) ≤ d ( n ) ≤ 2 Ω ( n ) . {\displaystyle 2^{\omega
Arithmetic_function
Spin model on a planar lattice
y_{p}\omega ^{j}},} where ω N = 1 {\displaystyle \omega ^{N}=1} is a primitive root of unity and we associate with each rapidity variable p three variables
Chiral_Potts_model
Concept in ring theory
F ( b ) {\displaystyle \chi _{n,F}(b)} . Then, since there exists a primitive root of unity ζ ∈ μ n ⊂ F {\displaystyle \zeta \in \mu _{n}\subset F} , there
Azumaya_algebra
Method for generating sequences of random integers
{\displaystyle 8k\pm 1} , b = 2 k {\displaystyle b=2^{k}} cannot be a primitive root of p = a b r − 1 {\displaystyle p=ab^{r}-1} . Therefore, MWC generators
Multiply-with-carry pseudorandom number generator
Multiply-with-carry_pseudorandom_number_generator
Polynomial equation of degree 3
by the primitive cube root of unity ε 1 = − 1 + i 3 2 , {\displaystyle \varepsilon _{1}={\frac {-1+i{\sqrt {3}}}{2}},} and the other cube root by the
Cubic_equation
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