scholarly journals On the quadratic form x 2 —7 y 2

1949 ◽  
Vol 197 (1049) ◽  
pp. 256-268 ◽  
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Binary Quadratic Form ◽  
Real Numbers ◽  
The Right ◽  
Indefinite Binary Quadratic Form

1. Let ƒ( x,y ) = ax 2 + bxy + cy 2 be an indefinite binary quadratic form, with real coefficients a, b, c , and discriminant d = b 2 —4 ac . A well-known theorem of Minkowski asserts that for any real numbers x 0 , y 0 there are integers x, y such that f(x + x0,y + yQ) (!) ♦Various authors (Heinhold 1938, 1939; Davenport 1946; Varnavides 1948 a ) have given results which are more precise than this. In particular, their results give, for a few special quadratic forms, the exact value of the constant, i.e. the least number by which ¼ √ d may be replaced on the right of (1). Perhaps the simplest form for which the existing results do not determine the exact value of the constant is the form x 2 — 7 y 2 with discriminant 28. Here ¼ √ d = ½ √7 and this was improved by Heinhold (1938, p. 683) to ¾. In this paper, we attack the problem by a method based on that of Davenport (1947), and find the exact value of the constant. We prove that it is always possible to satisfy

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).

The inhomogeneous minimum of binary quadratic forms

1992 ◽  
Vol 112 (1) ◽  
pp. 7-19 ◽  
Author(s):  
Daniel Berend ◽  
William Moran
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Binary Quadratic Form ◽  
Binary Quadratic Forms ◽  
General Behaviour ◽  
Inhomogeneous Minimum ◽  
The Subject ◽  
Indefinite Binary Quadratic Form

AbstractAn indefinite binary quadratic form ƒ gives rise to a certain function M on the torus. The properties of M, especially those related to its maximum – the so-called inhomogeneous minimum of ƒ – are the subject of numerous papers. Here we continue this study, putting more emphasis on the general behaviour of M.

The minima of a pair of indefinite, harmonic, binary quadratic forms

1945 ◽  
Vol 41 (2) ◽  
pp. 77-96 ◽  
Author(s):  
Kathleen Ollerenshaw
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Binary Quadratic Form ◽  
Binary Quadratic Forms ◽  
Indefinite Binary Quadratic Form

If f is a real, indefinite, binary quadratic form of discriminant d and if κ(f) is the minimum of | f | taken over all integer values of x, y, not both zero, then it is well known that and that this is a ‘best possible’ result.

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.

The inhomogeneous minimum of quadratic forms of signature ± 1

1981 ◽  
Vol 89 (2) ◽  
pp. 225-235 ◽  
Author(s):  
Madhu Raka
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Real Numbers ◽  
Inhomogeneous Minimum ◽  
Indefinite Quadratic Form ◽  
Indefinite Quadratic ◽  
Image Position

Let Qr be a real indefinite quadratic form in r variables of determinant D ≠ 0 and of type (r1, r2), 0 < r1 < r, r = r1 + r2, S = r1 − r2 being the signature of Qr. It is known (e.g. Blaney (3)) that, given any real numbers c1, c2,…, cr, there exists a constant C depending only on r and s such that the inequalityhas a solution in integers x1, x2, …, xr.

scholarly journals Positive values of inhomogeneous quinary quadratic forms of type (4,1)

1981 ◽  
Vol 31 (2) ◽  
pp. 175-188 ◽  
Author(s):  
R. J. Hans-Gill ◽  
Madhu Raka
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Real Numbers ◽  
Image Position

AbstractHere it is proved that if Q(x, y, z, t, u) is a real indefinite quinary quadratic form of type (4,1) and determinant D, then given any real numbers x0, y0, z0, t0, u0 there exist integers x, y, z, t, u such thatAll critical forms are also obtained.

Inhomogeneous minimum of indefinite quadratic forms in six variables: A conjecture of Watson

1983 ◽  
Vol 94 (1) ◽  
pp. 1-8 ◽  
Author(s):  
Madhu Raka
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Real Numbers ◽  
Famous Conjecture ◽  
Inhomogeneous Minimum ◽  
Indefinite Quadratic Form ◽  
History Of ◽  
Indefinite Quadratic ◽  
Image Position

The famous conjecture of Watson(11) on the minima of indefinite quadratic forms in n variables has been proved for n ≤ 5, n ≥ 21 and for signatures 0 and ± 1. For the details and history of the conjecture the reader is referred to the author's paper(8). In the succeeding paper (9), we prove Watson's conjecture for signature ± 2 and ± 3 and for all n. Thus only one case for n = 6 (i.e. forms of type (1, 5) or (5, 1)) remains to he proved which we do here; thereby completing the case n = 6. This result is also used in (9) for proving the conjecture for all quadratic forms of signature ± 4. More precisely, here we prove:Theorem 1. Let Q6(x1, …, x6) be a real indefinite quadratic form in six variables of determinant D ( < 0) and of type (5, 1) or (1, 5). Then given any real numbers ci, 1 ≤ i ≤ 6, there exist integers x1,…, x6such that

Quadratic forms over Quadratic Extensions of Fields with Two Quaternion Algebras

1979 ◽  
Vol 31 (5) ◽  
pp. 1047-1058 ◽  
Author(s):  
Craig M. Cordes ◽  
John R. Ramsey
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Square Root ◽  
Real Numbers ◽  
Quaternion Algebras ◽  
The Real ◽  
Quadratic Extensions

In this paper, we analyze what happens with respect to quadratic forms when a square root is adjoined to a field F which has exactly two quaternion algebras. There are many such fields—the real numbers and finite extensions of the p-adic numbers being two familiar examples. For general quadratic extensions, there are many unanswered questions concerning the quadratic form structure, but for these special fields we can clear up most of them.It is assumed char F ≠ 2 and K = F (√a) where a ∊ Ḟ – Ḟ2. Ḟ denotes the non-zero elements of F. Generally the letters a, b, c, … and α, β, … refer to elements from Ḟ and x, y, z, … come from .

scholarly journals Positive values of inhomogeneous 5-ary quadratic forms of type (3, 2)

1980 ◽  
Vol 29 (4) ◽  
pp. 439-453 ◽  
Author(s):  
R. J. Hans-Gill ◽  
Madhu Raka
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Real Numbers ◽  
Type 3

AbstractLet Q(x, y, z, t, u) be a real indefinite 5-ary quadratic form of type (3,2) and determinant D(> 0). Then given any real numbers x0, y0, z0, t0, u0 there exist integers x, y, z, t, u such that 0 < Q(x+x0,y+y0,z+z0,t+t0,u+u0)≦(16D)1/5. All the critical forms are also determined.

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