Growth of values of binary quadratic forms and Conway rivers

On the Gaps between Values of Binary Quadratic Forms

Among the values of a binary quadratic form, there are many twins of fixed distance. This is shown in quantitative form. For quadratic forms of discriminant −4 or 8 a corresponding result is obtained for triplets.

Applications of quadratic D-forms to generalized quadratic forms

In this paper, we study generalized quadratic forms over a division algebra with involution of the first kind in characteristic two. For this, we associate to every generalized quadratic from a quadratic form on its underlying vector space. It is shown that this form determines the isotropy behavior and the isometry class of generalized quadratic forms.

Composition of Binary Quadratic Forms

Division and Binary Quadratic Forms

has only three elements, written h(−23) = 3. There is an binary operation called composition that takes two primitive forms of the same discriminant to a third. Composition is commutative and associative, and makes the set of forms into a group, with identity 〈1, 0,−∆/4〉 for even discriminant and 〈1, 1, (1−∆)/4〉 for odd. From page 49 of Buell [1]: if a form 〈α, β, γ〉 represents a number r primi...

Powerfree Values of Binary Forms

Given a binary form f(x; y) 2 Z[x; y], we will be interested in nding the smallest k for which we can establish that there are in nitely many integers a and b such that f(a; b) is k free. Necessarily, we require the f has no xed kth prime power divisor. Until the nal section of this paper, we will also consider f to be irreducible. We set n = deg f . For k = 2, this problem has recently become ...