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) = ax2 + bxy + cy2 of discriminant Δ = b2 -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.

Authors:

• Ahmet Tekcan

Keywords:

• Quadratic form
• indefinite form
• cycle
• proper cycle
• right neighbor
• left neighbor.

References:

[1] J.Buchmann and U.Vollmer. Binary Quadratic Forms: An Algorithmic
Approach. Springer-Verlag, Berlin, Heidelberg, 2007.
[2] D.A.Buell. Binary Quadratic Forms, Clasical Theory and Modern Computations.
Springer-Verlag, New York, 1989.
[3] D.E.Flath. Introduction to Number Theory. Wiley, 1989.
[4] R.A. Mollin. Advanced Number Theory with Applications. CRC Press,
Taylor and Francis Group, Boca Raton, London, New York, 2009.
[5] A.Tekcan and O.Bizim. The Connection between Quadratic Forms and
the Extended Modular Group. Mathematica Bohemica 128(3)(2003), 225-
236.
[6] A.Tekcan. Proper Cycle of Indefinite Quadratic Forms and their Right
Neighbors. Applications of Mathematics 52(5)(2007), 407-415.
[7] A.Tekcan. A Second Approach to the Proper Cycles of Indefinite
Quadratic Forms and their Right Neighbors. Int. Journal of Contemporary
Math. Sci. 2(6)(2007), 249-260.