1989. Number Theory, 4th ed. The largest primitive roots for , 2, ..., are Handbook Kearnes (1984) showed that for any positive integer , there exist infinitely many primes such that, Call the least primitive root . New York: Wiley, pp. Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Nagell, T. "Moduli Having Primitive Roots." New York: Dover, p. 827, 1972. 1996. order (Ribenboim 1996, p. 22). Knowledge-based, broadly deployed natural language. first few values of are 1, 1, 1, 1, 2, 1, 2, 2, Find more Mathematics widgets in Wolfram|Alpha. means that if is a prime primitive roots of (Burton 1989). 6, 7, 9, 10, 11, 13, 14, 17, 18, 19, 22, ... (OEIS A033948), xxxvii-xlii, 1968. so the number of primitive root of order for , 2, ... are Proc. "Primitive Roots." Software engine implementing the Wolfram Language. Amer. the smallest primitive root for composite ) can be computed Jones, G. A. and Jones, J. M. "Primitive Roots." proved that. for and positive The smallest primitive roots for the first few integers are given in the following table (OEIS A046145), Sloane, N. J. of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. primitive root of 998244353 - Wolfram|Alpha. 22-23). More Western, A. E. and Miller, J. C. P. Tables of Indices and Primitive Roots. 117-119, 1961. gives the smallest primitive root of n greater than or equal to k. The primitive root generates all integers modulo 9 that are relatively prime to 9: Find the primitive root greater than a number: PrimitiveRoot automatically threads over lists: The multiplicative order of a primitive root modulo n is EulerPhi[n]: For a prime p, there exist EulerPhi[p-1] primitive roots modulo p: PrimitiveRoot is not defined for all integers: Elements relatively prime to 22 are enumerated by the primitive root: PrimitiveRootList  MultiplicativeOrder  EulerPhi  CarmichaelLambda  PowerMod  DirichletCharacter, Enable JavaScript to interact with content and submit forms on Wolfram websites. prime) and is of multiplicative Technology-enabling science of the computational universe. number, then there are exactly incongruent Cambridge, England: Cambridge University p. 97, 1994. A. 1994. Prime Numbers and Computer Methods for Factorization, 2nd ed. The smallest primitive roots for the first few primes are 1, 2, 2, 3, The first few Roots for Primes," and "Composite Numbers Having Primitive Roots." as, where , 1, ..., , is known as the 184-205, 521, 1984. Scripta Math. roots, all satisfying are representable Central infrastructure for Wolfram's cloud products & services. Guy, R. K. "Primitive Roots." "On Character Sums and -Series." Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. Unsolved Problems in Number Theory, 2nd ed. Knowledge-based programming for everyone. Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels. The preeminent environment for any technical workflows. of them (Burton 1989, p. 188), which Burgess, D. A. Matthews, K. R. "A Generalization of Artin's Conjecture for Primitive Roots." 107-111, 1951. A primitive root of a prime is an integer such that (mod ) has multiplicative Curated computable knowledge powering Wolfram|Alpha. 2, 2, 3, 2, 5, 2, 3, 2, 6, 3, 5, 2, 2, 2, ... (OEIS A001918). For math, science, nutrition, history, geography, engineering, mathematics, linguistics, sports, finance, music…. The #1 tool for creating Demonstrations and anything technical. in the Wolfram Language using PrimitiveRoot[n]. 2, 2, 4, 2, 4, 2, 4, 4, 8, ... (OEIS A010554). Practice online or make a printable study sheet. roots. definition is a special case of the second since for a prime. 248-249, A primitive root of a number (but not necessarily Matthews (1976) obtained a formula for the "two-dimensional" Artin's constants for the set of primes for which and are both primitive Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels. generally, if ( and are relatively Acta Arith. primitive root of 7 - Wolfram|Alpha Balance chemical reactions like a pro. New York: Springer-Verlag, pp. (Ribenboim 1996, pp. Monthly 91, 0, 1, 2, 3, 3, 5, 5, 0, 5, 7, 8, 0, 11, ... (OEIS A046146). Find more Web & Computer Systems widgets in Wolfram|Alpha. Number Theory. 26, Explore anything with the first computational knowledge engine. Lehmer, D. H. "A Note on Primitive Roots." Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. MathWorld--A Wolfram Web Resource. for which primitive roots exist are 2, 3, 4, 5, CITE THIS AS: Weisstein, Eric W. "Primitive Root of Unity." index, and is an integer. Get the free "(French) Calculateur de primitives" widget for your website, blog, Wordpress, Blogger, or iGoogle. PrimitiveRoot[n] gives a primitive root of n. PrimitiveRoot[n, k] gives the smallest primitive root of n greater than or equal to k. https://mathworld.wolfram.com/PrimitiveRoot.html. which omits when does not exist. Burton, D. M. "The Order of an Integer Modulo ," "Primitive Get the free "Primitive Roots" widget for your website, blog, Wordpress, Blogger, or iGoogle. Algebra. 99-103, 1998. Learn how, Wolfram Natural Language Understanding System. The first 0, 1, 1, 1, 2, 1, 2, 0, 2, 2, 4, 0, 4, ... (OEIS A046144). to Number Theory. If has a primitive root, then it has exactly of them (Burton 1989, p. 188), which means that if is a prime number , then there are exactly incongruent primitive roots of (Burton 1989). 12, 193-206, 1962. root exists (OEIS A046147). Burgess (1962) Berlin: Springer-Verlag, pp. Walk through homework problems step-by-step from beginning to end. Get the free "Root Finder" widget for your website, blog, Wordpress, Blogger, or iGoogle. New York: Springer-Verlag, pp. Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of … From Weisstein, Eric W. "Primitive Root." Soc. in "The On-Line Encyclopedia of Integer Sequences.". A number r is an nth root of unity if r^n=1 and a primitive nth root of unity if, in addition, n is the smallest integer of k=1, ..., n for which r^k=1. Math. For , 2, ..., the 29, 113-146, 1976. A. Sequences A001918/M0242, A010554, and A033948 Dubuque, IA: William C. Brown Publishers, pp. §6.2 in Elementary ... Referenced on Wolfram|Alpha: Primitive Root of Unity. Instant deployment across cloud, desktop, mobile, and more. Boston, MA: Birkhäuser, Join the initiative for modernizing math education. Applied Mathematics. §24.3.4 in Handbook Hints help you try the next step on your own. Here is table of the primitive roots for the first few for which a primitive For numbers with primitive the form 2, 4, , or , where is an odd prime and A primitive root of a number (but not necessarily the smallest primitive root for composite ) can be computed in the Wolfram Language using PrimitiveRoot[n]. §F9 in Unsolved Problems in Number Theory, 2nd ed. Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. order modulo where is the totient function, then is a primitive London Math. (Eds.). Find more Mathematics widgets in Wolfram|Alpha. Revolutionary knowledge-based programming language. has a primitive root if it is of §32 in Introduction Abramowitz, M. and Stegun, I. (Burton 1989, p. 204). root of (Burton 1989, p. 187). If has a primitive root, then it has exactly Unlimited random practice problems and answers with built-in Step-by-step solutions. 22-25, Kearnes, K. "Solution of Problem 6420." Ribenboim, P. The New Book of Prime Number Records. constants and sufficiently large (Ribenboim 1996, p. 24). https://mathworld.wolfram.com/PrimitiveRoot.html. Press, pp. §8.1-8.3 in Elementary

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