Dictionary

primitive root en · NOUN

Meanings

1. For a given modulus n, a number g such that for every a coprime to n there exists an integer k such that gᵏ ≡ a (mod n); a generator (or primitive element) of the multiplicative group, modulo n, of integers relatively prime to n.
   • The integers 2, 3, 4, and 6 each have exactly one primitive root and therefore, by default, each has a set of primitive roots consisting of "consecutive" integers. The integer 5, with primitive roots of 2 and 3 is the only positive integer having at least two primitive roots for which the entire set of primitive roots are consecutive integers. 1992, Joe Roberts, Lure of the Integers, Mathematical Association of America, page 55:
   • There are #92;phi(p-1) incongruent primitive roots of p. The fact that there are so many primitive roots causes no difficulty in the theory of the binomial congruence but has caused considerable confusion in the tabulation of primitive roots. 1941, Derrick Henry Lehmer, Guide to Tables in the Theory of Numbers, National Research Council, page 13:
   • For example, the prime 7 has #92;phi(6)#61;2 primitive roots, namely, 3 and 5. Also, the prime 11 has #92;phi(10)#61;4 primitive roots, namely, 2, 6, 7, 8. Recall from Theorem 6.7 that if m has primitive roots, and if g is one primitive root (#92;operatorname#123;mod#125;m), then we can obtain all primitive roots (#92;operatorname#123;mod#125;m) by raising g to appropriate exponents. 2006, Neville Robbins, Beginning Number Theory, Jones & Bartlett Learning, page 159:

Forms

Relateds

multiplicative order

Synonyms

primitive element (number that generates other numbers modulo n) · generator (number that generates other numbers modulo n)

Translations (7)

it radice primitiva (number such that gk ≡ a (mod n) exists for every a coprime to n — see also generator, primitive element) · de Primitivwurzel (number such that gk ≡ a (mod n) exists for every a coprime to n — see also generator, primitive element) · fi primitiivinen juuri (number such that gk ≡ a (mod n) exists for every a coprime to n — see also generator, primitive element) · is frumstæð rót (number such that gk ≡ a (mod n) exists for every a coprime to n — see also generator, primitive element) · fr racine primitive (number such that gk ≡ a (mod n) exists for every a coprime to n — see also generator, primitive element) · pl pierwiastek pierwotny (number such that gk ≡ a (mod n) exists for every a coprime to n — see also generator, primitive element) · cmn 原根 (number such that gk ≡ a (mod n) exists for every a coprime to n — see also generator, primitive element)