For example x 2 82y 2 and 2x 2 41y 2 are in the same genus but not equivalent over Z.

Example of binary quadratic forms. The references are Hec10 Ch. Forms in the same genus have equal discriminant and hence there are only finitely many equivalence classes in a genus. Given an integer n we can also ask which binary quadratic forms could represent n.

Every positive prime p 1 mod 4 can be written in the form p x2 y2. 89 509 509 89 64 89 2 89 8 1 since 1 1. Let p be a prime number.

That is there is exactly one reduced binary quadratic forms of discriminant 4 namely x2 y2 and hence just one equivalence class of binary quadratic forms of discriminant. 3 Binary Quadratic Forms At the end of this section we will be able to use the theory of reduced forms to characterize primes of the form p x2 ny2 for n 12347. General number fields and quadratic forms in n variables.

For example x2 y2 is a BQF x2 13y2 is a BQF 2x2 5xy 17y2 is a BQF. I If p am2 bmn cn2 for some integers mn then dis a square mod 4p. Given a binary quadratic form fxy ax2 bxycy2 we can ask which integers n can be written as n fx 0y 0forsomex 0y 0 in which case f is said to represent n.

Some of the brightest mathematicians have contributed to this theory since then including Euler Lagrange Gauß and many others. For binary quadratic forms there is a group structure on the set C equivalence classes of forms with given discriminant. R x y ZZ sage.

A binary quadratic form is written a b c and refers to the expression a x2 b x y c y2. The binary quadratic form ax2 bxy cy2. The associated symmetric matrix M f so that fxy xy M f x y is M f a b2 b2 c.