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# State/Arithmetic progression rule for binary quadratic forms

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