The structure of the number of representations function in a positive binary quadratic form

1933 ◽  
Vol 36 (1) ◽  
pp. 321-343 ◽  
Author(s):  
Gordon Pall
Keyword(s):  
Quadratic Form ◽  
Binary Quadratic Form ◽  
Positive Binary Quadratic Form

scholarly journals An Inhomogeneous Minimum For Non-Convex Star-Regions With Hexagonal Symmetry

1955 ◽  
Vol 7 ◽  
pp. 337-346 ◽  
Author(s):  
R. P. Bambah ◽  
K. Rogers
Keyword(s):  
Quadratic Form ◽  
Binary Quadratic Form ◽  
Elementary Geometry ◽  
Hexagonal Symmetry ◽  
Real Numbers ◽  
Inhomogeneous Minimum ◽  
Image Position ◽  
Indefinite Binary Quadratic Form

1. Introduction. Several authors have proved theorems of the following type:Let x0, y0 be any real numbers. Then for certain functions f(x, y), there exist numbers x, y such that1.1 x ≡ x0, y ≡ y0 (mod 1),and1.2 .The first result of this type, but with replaced by min , was given by Barnes (3) for the case when the function is an indefinite binary quadratic form. A generalisation of this was proved by elementary geometry by K. Rogers (6).

scholarly journals Minimal theta functions

1988 ◽  
Vol 30 (1) ◽  
pp. 75-85 ◽  
Author(s):  
Hugh L. Montgomery
Keyword(s):  
Functional Equation ◽  
Quadratic Form ◽  
Zeta Function ◽  
Theta Functions ◽  
Binary Quadratic Form ◽  
Positive Definite ◽  
Epstein Zeta Function ◽  
Image Position

Let be a positive definite binary quadratic form with real coefficients and discriminant b2 − 4ac = −1.Among such forms, let . The Epstein zeta function of f is denned to beRankin [7], Cassels [1], Ennola [5], and Diananda [4] between them proved that for every real s > 0,We prove a corresponding result for theta functions. For real α > 0, letThis function satisfies the functional equation(This may be proved by using the formula (4) below, and then twice applying the identity (8).)

Valuation Rings and Rigid Elements in Fields

1981 ◽  
Vol 33 (6) ◽  
pp. 1338-1355 ◽  
Author(s):  
Roger Ware
Keyword(s):  
Quadratic Form ◽  
Nonzero Element ◽  
Discrete Valuation Ring ◽  
Binary Quadratic Form ◽  
Valuation Rings ◽  
Unique Decomposition ◽  
Complete Discrete Valuation Ring ◽  
Residue Class Field ◽  
Uniformizing Parameter ◽  
Anisotropic Quadratic Form

In [20], T. A. Springer proved that if A is a complete discrete valuation ring with field of fractions F, residue class field of characteristic not 2, and uniformizing parameter π then any anisotropic quadratic form q over F has a unique decomposition as q = q1 ⊥ 〈π〉q2, where q1 and q2 represent only units of A, modulo squares in F (compare [14, Satz 12.2.2], [19, §4], [18, Theorem 8.9]). Consequently the binary quadratic form x2 + πy2 represents only elements in Ḟ2 ∪ πḞ2, where Ḟ2 denotes the set of nonzero squares in F. Szymiczek [21] has called a nonzero element a in a field F rigid if the binary quadratic form x2 + ay2 represents only elements in Ḟ2 ∪ aḞ2.

An Elementary Proof of a Theorem About the Representation of Primes by Quadratic Forms

1954 ◽  
Vol 6 ◽  
pp. 353-363 ◽  
Author(s):  
W. E. Briggs
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Elementary Proof ◽  
Prime Numbers ◽  
Binary Quadratic Form ◽  
First Case

The theorem that every properly primitive binary quadratic form is capable of representing infinitely many prime numbers was first proved completely by H. Weber (5). The purpose of this paper is to give an elementary proof of the case where the form is ax2 + 2bxy + cy2, with a > 0, (a, 2b, c) = 1, and D = b2 — ac not a square. The cases where the form is ax2 + bxy + cy2 with b odd, and the case where the form is ax2+ 2bxy + cy2 with D a square, can be settled very simply once the first case is taken care of, and this is done in a page and a half in the Weber paper.

Quartic residuacity and the quadratic character of certain quadratic irrationalities

2015 ◽  
Vol 11 (08) ◽  
pp. 2487-2503
Author(s):  
Ronald J. Evans ◽  
Kenneth S. Williams
Keyword(s):  
Quadratic Form ◽  
General Theorem ◽  
Binary Quadratic Form ◽  
Legendre Symbol ◽  
Quadratic Character

We prove a general theorem that evaluates the Legendre symbol [Formula: see text] under certain conditions on the integers A, B, m and the prime p. The evaluation is in terms of parameters appearing in a binary quadratic form representing p. The theorem has applications to quartic residuacity.

REPRESENTATIONS OF INTEGERS BY THE BINARY QUADRATIC FORM

2015 ◽  
Vol 100 (2) ◽  
pp. 182-191 ◽  
Author(s):  
BUMKYU CHO
Keyword(s):  
Field Theory ◽  
Quadratic Form ◽  
Binary Quadratic Form ◽  
Class Field Theory ◽  
Class Numbers ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Class Field ◽  
Representations Of Integers ◽  
Necessary And Sufficient

In terms of class field theory we give a necessary and sufficient condition for an integer to be representable by the quadratic form $x^{2}+xy+ny^{2}$ ($n\in \mathbb{N}$ arbitrary) under extra conditions $x\equiv 1\;\text{mod}\;m$, $y\equiv 0\;\text{mod}\;m$ on the variables. We also give some examples where their extended ring class numbers are less than or equal to $3$.

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