Displaying similar documents to “Representation numbers of five sextenary quadratic forms”

On the number of representations of a positive integer by certain quadratic forms

Ernest X. W. Xia (2014)

Colloquium Mathematicae

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For natural numbers a,b and positive integer n, let R(a,b;n) denote the number of representations of n in the form i = 1 a ( x ² i + x i y i + y ² i ) + 2 j = 1 b ( u ² j + u j v j + v ² j ) . Lomadze discovered a formula for R(6,0;n). Explicit formulas for R(1,5;n), R(2,4;n), R(3,3;n), R(4,2;n) and R(5,1;n) are determined in this paper by using the (p;k)-parametrization of theta functions due to Alaca, Alaca and Williams.

Sumsets in quadratic residues

I. D. Shkredov (2014)

Acta Arithmetica

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We describe all sets A p which represent the quadratic residues R p in the sense that R = A + A or R = A ⨣ A. Also, we consider the case of an approximate equality R ≈ A + A and R ≈ A ⨣ A and prove that A is then close to a perfect difference set.

Positivity of quadratic base change L -functions

Hervé Jacquet, Chen Nan (2001)

Bulletin de la Société Mathématique de France

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We show that certain quadratic base change L -functions for Gl ( 2 ) are non-negative at their center of symmetry.

Another look at real quadratic fields of relative class number 1

Debopam Chakraborty, Anupam Saikia (2014)

Acta Arithmetica

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The relative class number H d ( f ) of a real quadratic field K = ℚ (√m) of discriminant d is defined to be the ratio of the class numbers of f and K , where K denotes the ring of integers of K and f is the order of conductor f given by + f K . R. Mollin has shown recently that almost all real quadratic fields have relative class number 1 for some conductor. In this paper we give a characterization of real quadratic fields with relative class number 1 through an elementary approach considering the...

Capturing forms in dense subsets of finite fields

Brandon Hanson (2013)

Acta Arithmetica

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An open problem of arithmetic Ramsey theory asks if given an r-colouring c:ℕ → 1,...,r of the natural numbers, there exist x,y ∈ ℕ such that c(xy) = c(x+y) apart from the trivial solution x = y = 2. More generally, one could replace x+y with a binary linear form and xy with a binary quadratic form. In this paper we examine the analogous problem in a finite field q . Specifically, given a linear form L and a quadratic form Q in two variables, we provide estimates on the necessary size...

On geodesics of phyllotaxis

Roland Bacher (2014)

Confluentes Mathematici

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Seeds of sunflowers are often modelled by n ϕ θ ( n ) = n e 2 i π n θ leading to a roughly uniform repartition with seeds indexed by consecutive integers at angular distance 2 π θ for θ the golden ratio. We associate to such a map ϕ θ a geodesic path γ θ : > 0 PSL 2 ( ) of the modular curve and use it for local descriptions of the image ϕ θ ( ) of the phyllotactic map ϕ θ .

Minimal 𝒮 -universality criteria may vary in size

Noam D. Elkies, Daniel M. Kane, Scott Duke Kominers (2013)

Journal de Théorie des Nombres de Bordeaux

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In this note, we give simple examples of sets 𝒮 of quadratic forms that have minimal 𝒮 -universality criteria of multiple cardinalities. This answers a question of Kim, Kim, and Oh [KKO05] in the negative.

On some universal sums of generalized polygonal numbers

Fan Ge, Zhi-Wei Sun (2016)

Colloquium Mathematicae

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For m = 3,4,... those pₘ(x) = (m-2)x(x-1)/2 + x with x ∈ ℤ are called generalized m-gonal numbers. Sun (2015) studied for what values of positive integers a,b,c the sum ap₅ + bp₅ + cp₅ is universal over ℤ (i.e., any n ∈ ℕ = 0,1,2,... has the form ap₅(x) + bp₅(y) + cp₅(z) with x,y,z ∈ ℤ). We prove that p₅ + bp₅ + 3p₅ (b = 1,2,3,4,9) and p₅ + 2p₅ + 6p₅ are universal over ℤ, as conjectured by Sun. Sun also conjectured that any n ∈ ℕ can be written as p ( x ) + p ( y ) + p 11 ( z ) and 3p₃(x) + p₅(y) + p₇(z) with x,y,z...

Tameness criterion for posets with zero-relations and three-partite subamalgams of tiled orders

Stanisław Kasjan (2002)

Colloquium Mathematicae

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A criterion for tame prinjective type for a class of posets with zero-relations is given in terms of the associated prinjective Tits quadratic form and a list of hypercritical posets. A consequence of this result is that if Λ is a three-partite subamalgam of a tiled order then it is of tame lattice type if and only if the reduced Tits quadratic form q Λ associated with Λ in [26] is weakly non-negative. The result generalizes a criterion for tameness of such orders given by Simson [28]...

Grothendieck and Witt groups in the reduced theory of quadratic forms

Andrzej Sładek (1980)

Annales Polonici Mathematici

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Abstract. Let F be a formally real field. Denote by G(F) and G t ( F ) the Grothen-dieck group of quadratic forms over F and its torsion subgroup, respectively. In this paper we study the structure of the factor group G ( F ) / G t ( F ) . This reduced Grothendieck group is a free Abelian group. The main results of the paper describe some sets of generators for G ( F ) / G t ( F ) , which in many cases allow us to find a basis for the group. Throughout the paper we use the language of the reduced theory of quadratic forms. In the...

Sums of Three Prime Squares

Hiroshi Mikawa, Temenoujka Peneva (2007)

Bollettino dell'Unione Matematica Italiana

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Let A , ϵ > 0 be arbitrary. Suppose that x is a sufficiently large positive number. We prove that the number of integers n ( x , x + x θ ] , satisfying some natural congruence conditions, which cannot be written as the sum of three squares of primes is x θ ( log x ) - A , provided that 7 16 + ϵ θ 1 .

The sum of divisors of a quadratic form

Lilu Zhao (2014)

Acta Arithmetica

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We study the sum τ of divisors of the quadratic form m₁² + m₂² + m₃². Let S ( X ) = 1 m , m , m X τ ( m ² + m ² + m ² ) . We obtain the asymptotic formula S₃(X) = C₁X³logX + C₂X³ + O(X²log⁷X), where C₁,C₂ are two constants. This improves upon the error term O ε ( X 8 / 3 + ε ) obtained by Guo and Zhai (2012).