Binary quadratic form: Difference between revisions

Revision as of 19:18, 27 November 2017

Revision as of 19:40, 2 July 2018

Line 140:

Suppose we wish to compose forms <math>f_1 = A_1 x^2 + B_1 xy + C_1 y^2</math> and <math>f_2 = A_2 x^2 + B_2 xy + C_2 y^2</math>, each primitive and of the same discriminant <math>\Delta</math>. We perform the following steps:

# Compute <math>BB_\mu = \tfrac{B_1 + B_2}{2}</math> and <math> e = \gcd(A_1, A_2, BB_\mu)</math>, and <math>A = \tfrac{A_1 A_2}{e^2}</math>

# Solve the system of congruences <blockquote><math>\begin{align} x &\equiv B_1 \pmod{2 \tfrac{A_1}{e}}\\ x &\equiv B_2 \pmod{2 \tfrac{A_2}{e}}\\ \tfrac{BB_\mu}{e} x &\equiv \tfrac{\Delta + B_1 B_2}{2e} \pmod{2A} \end{align} </math></blockquote>It can be shown that this system always has a unique integer solution modulo <math>2A</math>. We arbitrarily choose such a solution and call it ''B''.

# Compute ''C'' such that <math>\Delta = B^2 - 4AC</math>. It can be shown that ''C'' is an integer.