Norm Forms Represent Few Integers but Relatively Many Primes
نویسندگان
چکیده
are integral forms arising from norms on number fields. We prove that the natural density of the set of integers represented by a norm form is zero, while the relative natural density of the set of prime numbers represented by a norm form exists and is positive. These results require tools from class field theory, including the Artin-Chebotarev density theorem and the Hilbert class field. We introduce these tools as we need them in the course of the main arguments. This article is expository in nature and assumes only a first course in algebraic number theory.
منابع مشابه
How infinite primes relate to other views about mathematical infinity?
Infinite primes is a purely TGD inspired notion. The notion of infinity is number theoretical and infinite primes have well defined divisibility properties. One can partially order them by the real norm. p-Adic norms of infinite primes are well defined and finite. The construction of infinite primes is a hierarchical procedure structurally equivalent to a repeated second quantization of a super...
متن کاملThe power digraphs of safe primes
A power digraph, denoted by $G(n,k)$, is a directed graph with $Z_{n}={0,1,..., n-1}$ as the set of vertices and $L={(x,y):x^{k}equiv y~(bmod , n)}$ as the edge set, where $n$ and $k$ are any positive integers. In this paper, the structure of $G(2q+1,k)$, where $q$ is a Sophie Germain prime is investigated. The primality tests for the integers of the form $n=2q+1$ are established in terms of th...
متن کاملSmall gaps between almost primes, the parity problem and some conjectures of Erdős on consecutive integers
In a previous paper, the authors proved that in any system of three linear forms satisfying obvious necessary local conditions, there are at least two forms that infinitely often assume E2-values; i.e., values that are products of exactly two primes. We use that result to prove that there are inifinitely many integers x that simultaneously satisfy ω(x) = ω(x + 1) = 4,Ω(x) = Ω(x + 1) = 5, and d(...
متن کامل8430 Handout 3: Elementary Theory of Quadratic Forms
Let f = ax + bxy+ cy be a quadratic form and suppose that d = gcd(a, b, c) > 1. Then f can only represent integers which are multiples of d – in particular it can represent at most one prime, and no primes at all if d is not prime. Moreover f represents dN iff the integral form 1 df represents N , so there is no loss of generality in considering only the case gcd(a, b, c) = 1. Such forms are sa...
متن کاملRepresentation Functions of Bases for Binary Linear Forms
Let F (x1, . . . , xm) = u1x1 + · · · + umxm be a linear form with relatively prime integer coefficients u1, . . . , um. For any set A of integers, let F (A) = {F (a1, . . . , am) : ai ∈ A for i = 1, . . . , m}. The representation function associated with the form F is RA,F (n) = card ({(a1, . . . , am) ∈ A m : F (a1, . . . , am) = n}) . The set A is an asymptotic basis with respect to F if RA,...
متن کامل