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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.
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.
In computational number theory, the Lucas test is a primality test for a natural number n; it requires that the prime factors of n − 1 be already known. [ 1 ] [ 2 ] It is the basis of the Pratt certificate that gives a concise verification that n is prime.
Let a be an integer that is not a square number and not −1. Write a = a 0 b 2 with a 0 square-free. Denote by S(a) the set of prime numbers p such that a is a primitive root modulo p. Then the conjecture states S(a) has a positive asymptotic density inside the set of primes. In particular, S(a) is infinite.
If the modulus and conductor are equal the character is primitive, otherwise imprimitive. An imprimitive character is induced by the character for the smallest modulus: χ 16 , 9 {\displaystyle \chi _{16,9}} is induced from χ 8 , 5 {\displaystyle \chi _{8,5}} and χ 16 , 15 {\displaystyle \chi _{16,15}} and χ 8 , 7 {\displaystyle \chi _{8,7 ...
Prime-counting function: Number of primes less than or equal to a given number. Partition function: Order-independent count of ways to write a given positive integer as a sum of positive integers. Möbius μ function: Sum of the nth primitive roots of unity, it depends on the prime factorization of n. Prime omega functions; Chebyshev functions
A former Tennessee teacher who got pregnant after raping a 12-year-old boy pleaded guilty and has been sentenced to 25 years in prison with no parole. On Dec. 20, Alissa McCommon, 39, of Covington ...
For a primitive () th root x, the number () / is a primitive th root of unity. If k does not divide λ ( n ) {\displaystyle \lambda (n)} , then there will be no k th roots of unity, at all. Finding multiple primitive k th roots modulo n