State/Arithmetic progression rule for binary quadratic forms

From SlugmathWiki

Proposition
Proof

Proposition: (Arithmetic progression rule for binary quadratic forms) Suppose that $\{ \pm \vec e, \pm \vec f \}$ is a lax basis. Suppose that $Q(x,y) = ax^2 + bxy + cy^2$ is an integer-valued binary quadratic form.

Then, the following three integers are in arithmetic progression: $$Q(\vec e - \vec f), Q(\vec e) + Q(\vec f), Q(\vec e + \vec f).$$

Logical Connections

This statement logically relies on the following definitions and statements: Def/Lax basis, Def/Binary quadratic form, Def/Arithmetic progression

The following statements and definitions rely on the material of this page: State/Binary quadratic forms are uniquely determined by their values at a superbasis, State/Bounding riverbends of a given discriminant, State/Climbing in topographs, State/The domain topograph has no circuits, and State/Values in the range topograph adjacent to a given face form a quadratic sequence

To visualize the logical connections between this statements and other items of mathematical knowledge, you can visit the following cluster(s), and click the "Visualize" tab: Clust/Binary quadratic forms

State/Arithmetic progression rule for binary quadratic forms

From SlugmathWiki

Logical Connections

Views

Personal tools

Navigation

Mathematics

Pegagogy

Search

Toolbox

	Suppose that $\{ \pm \vec e, \pm \vec f \}$ is a lax basis. Suppose that $Q(x,y) = ax^2 + bxy + cy^2$ is an integer-valued binary quadratic form. Suppose that $\vec e = (u,v)$ and $\vec f = (r,s)$.
	We evaluate the three quantities of interest: $Q(\vec e - \vec f) = a (u-r)^2 + b(u-r)(v-s) + c(v-s)^2$. $Q(\vec e) + Q(\vec f) = au^2 + buv + cv^2 + ar^2 + brs + cs^2$. $Q(\vec e + \vec f) = a (u+r)^2 + b(u+r)(v+s) + c(v+s)^2$. Direct computation demonstrates that this sequence of integers has common difference $D$ given by: $$D = 2aur + bus + brv + 2cvs.$$
	Hence the three integers are in arithmetic progression.