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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.
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:
Jacobi's original tables use 10 or −10 or a number with a small power of this form as the primitive root whenever possible, while the second edition uses the smallest possible positive primitive root (Fletcher 1958). The term "canon arithmeticus" is occasionally used to mean any table of indices and powers of primitive roots.
If x is a primitive kth root of unity and also a (not necessarily primitive) ℓth root of unity, then k is a divisor of ℓ. This is true, because Bézout's identity yields an integer linear combination of k and ℓ equal to gcd ( k , ℓ ) {\displaystyle \gcd(k,\ell )} .
Toggle the table of contents. Primitive root. ... Printable version; In other projects ... In mathematics, a primitive root may mean: Primitive root modulo n in ...
Toggle the table of contents. ... Print/export Download as PDF; Printable version; ... −1 is a primitive root mod 4 ...
It may also be defined as the monic polynomial with integer coefficients that is the minimal polynomial over the field of the rational numbers of any primitive nth-root of unity (/ is an example of such a root). An important relation linking cyclotomic polynomials and primitive roots of unity is
For these primes, 2 is a primitive root modulo p, so 2 n modulo p can be any natural number between 1 and p − 1. a ( i ) = 2 i mod p mod 2 . {\displaystyle a(i)=2^{i}{\bmod {p}}{\bmod {2}}.} These sequences of period p − 1 have an autocorrelation function that has a negative peak of −1 for shift of ( p − 1 ) / 2 {\displaystyle (p-1)/2} .