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For example, 20 is a primitive abundant number because: The sum of its proper divisors is 1 + 2 + 4 + 5 + 10 = 22, so 20 is an abundant number. The sums of the proper divisors of 1, 2, 4, 5 and 10 are 0, 1, 3, 1 and 8 respectively, so each of these numbers is a deficient number. The first few primitive abundant numbers are:
A semiperfect number that is not divisible by any smaller semiperfect number is called primitive. Every number of the form 2 m p for a natural number m and an odd prime number p such that p < 2 m+1 is also semiperfect. In particular, every number of the form 2 m (2 m+1 − 1) is semiperfect, and indeed perfect if 2 m+1 − 1 is a Mersenne prime.
Naive set theory: The empty set is a primitive notion. To assert that it exists would be an implicit axiom. Peano arithmetic: The successor function and the number zero are primitive notions. Since Peano arithmetic is useful in regards to properties of the numbers, the objects that the primitive notions represent may not strictly matter. [7]
In number theory, a kth root of unity modulo n for positive integers k, n ≥ 2, is a root of unity in the ring of integers modulo n; that is, a solution x to the equation (or congruence) (). If k is the smallest such exponent for x, then x is called a primitive kth root of unity modulo n. [1]
A primitive polynomial must have a non-zero constant term, for otherwise it will be divisible by x. Over GF(2), x + 1 is a primitive polynomial and all other primitive polynomials have an odd number of terms, since any polynomial mod 2 with an even number of terms is divisible by x + 1 (it has 1 as a root).
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.
A Fermat number cannot be a perfect number or part of a pair of amicable numbers. The series of reciprocals of all prime divisors of Fermat numbers is convergent. (Křížek, Luca & Somer 2002) If n n + 1 is prime, there exists an integer m such that n = 2 2 m. The equation n n + 1 = F (2 m +m) holds in that case. [13] [14]
In number theory, Artin's conjecture on primitive roots states that a given integer a that is neither a square number nor −1 is a primitive root modulo infinitely many primes p. The conjecture also ascribes an asymptotic density to these primes. This conjectural density equals Artin's constant or a rational multiple thereof.