Neighbors of Indefinite Binary Quadratic Forms

In this paper, we derive some algebraic identities on right and left neighbors R(F ) and L(F ) of an indefinite binary quadratic form F = F (x, y) = ax + bxy + cy of discriminant Δ = b − 4ac. We prove that the proper cycle of F can be given by using its consecutive left neighbors. Also we construct a connection between right and left neighbors of F . Keywords—Quadratic form, indefinite form, cycle, proper cycle, right neighbor, left neighbor. I. PRELIMINARIES. A real binary quadratic form F is a polynomial in two variables x and y of the type F = F (x, y) = ax + bxy + cy (1) with real coefficients a, b, c. We denote it by F = (a, b, c). The discriminant of F is defined by the formula b2− 4ac and is denoted by Δ = Δ(F ). F is an integral form if and only if a, b, c ∈ Z, and is called indefinite if and only if Δ(F ) > 0. An indefinite form F = (a, b, c) of discriminant Δ is said to be reduced if ∣∣∣√Δ− 2|a|∣∣∣ < b < Δ. (2) Most properties of quadratic forms can be giving by the aid of extended modular group Γ (see [5]). Gauss (1777-1855) defined the group action of Γ on the set of forms as follows: gF (x, y) = ( ar + brs+ cs ) x +(2art+ bru+ bts+ 2csu)xy (3) + ( at + btu+ cu ) y for g = ( r s t u ) ∈ Γ. Hence two forms F and G are called equivalent if and only if there exists a g ∈ Γ such that gF = G. If det g = 1, then F and G are called properly equivalent, and if det g = −1, then F and G are called improperly equivalent. If a form F is improperly equivalent to itself, then it called ambiguous. Let ρ(F ) denotes the normalization (it means that replacing F by its normalization) of (c,−b, a). To be more explicit, we set ρ(F ) = (c,−b+ 2cri, cr i − bri + a), (4)