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 modulon if for every integer a coprime to n, there is some integer k for which gk ≡ a (mod n). Such a value k is called the index or discrete logarithm of a to the base g modulo n. So g is a primitive root modulon if and only if g is a generator of the multiplicative group of integers modulo n.
Gauss defined primitive roots in Article 57 of the Disquisitiones Arithmeticae (1801), where he credited Euler with coining the term. In Article 56 he stated that Lambert and Euler knew of them, but he was the first to rigorously demonstrate that primitive roots exist for a prime n. In fact, the Disquisitiones contains two proofs: The one in Article 54 is a nonconstructive existence proof, while the proof in Article 55 is constructive.
A primitive root exists if and only if n is 1, 2, 4, pk or 2pk, where p is an odd prime and k > 0. For all other values of n the multiplicative group of integers modulo n is not cyclic.[1][2][3]
This was first proved by Gauss.[4]
^Weisstein, Eric W. "Modulo Multiplication Group". MathWorld.
^Primitive root, Encyclopedia of Mathematics
^(Vinogradov 2003, pp. 105–121, § VI PRIMITIVE ROOTS AND INDICES)
once). A generator of ( Z / n Z ) × {\displaystyle (\mathbb {Z} /n\mathbb {Z} )^{\times }} is called a primitiverootmodulon. If there is any generator...
exponentiation Modulo (mathematics) Multiplicative group of integers modulon Pisano period (Fibonacci sequences modulon) Primitiverootmodulon Quadratic...
In mathematics, the term primitive element can mean: Primitiverootmodulon, in number theory Primitive element (field theory), an element that generates...
an element whose order equals the exponent, λ(n). Such an element is called a primitive λ-rootmodulon. The Carmichael function is named after the American...
of modular integers, see Root of unity modulon. Every nth root of unity z is a primitive ath root of unity for some a ≤ n, which is the smallest positive...
multiplier a is an element of high multiplicative order modulo m (e.g., a primitiverootmodulon), and the seed X0 is coprime to m. Other names are multiplicative...
\Phi _{n}} is irreducible if and only if p is a primitiverootmodulon, that is, p does not divide n, and its multiplicative order modulon is φ ( n ) {\displaystyle...
and Mangerel. Character sum Multiplicative group of integers modulonPrimitiverootmodulon Multiplicative character This is the standard definition; e...
quadratic residue modulon if it is congruent to a perfect square modulon; i.e., if there exists an integer x such that: x 2 ≡ q ( mod n ) . {\displaystyle...
contains a nth primitiveroot 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...
integer c is the hypotenuse of a primitive Pythagorean triple if and only if each prime factor of c is congruent to 1 modulo 4; that is, each prime factor...
prime, then there exists a primitiverootmodulon, or generator of the group (Z/nZ)*. Such a generator has order |(Z/nZ)*| = n−1 and both equivalences will...
alternating sum of digits yields the value modulo ( b + 1 ) {\displaystyle (b+1)} . It helps to see the digital root of a positive integer as the position...
every primitiven-th root of unity is also a principal n{\displaystyle n}-th root of unity. In any ring, if n is a power of 2, then any n/2-th root of −1...
0x80) /* GF modulo: if a has a nonzero term x^7, then must be reduced when it becomes x^8 */ a = (a << 1) ^ 0x11b; /* subtract (XOR) the primitive polynomial...
multiplicative order ordp b = p − 1, which is equivalent to b being a primitiverootmodulo p. The term "long prime" was used by John Conway and Richard Guy...
the rational root test. If the polynomial to be factored is a n x n + a n − 1 x n − 1 + ⋯ + a 1 x + a 0 {\displaystyle a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}}...
integers. Every finite cyclic group of order n is isomorphic to the additive group of Z/nZ, the integers modulon. Every cyclic group is an abelian group (meaning...
Global Information