Zeros Of Binary Quadratic Forms Modulo N And Class Numbers Of Imaginary Quadratic Fields

In this paper,the main object we study is the binary quadratic form f(x,y)?ax2 + bxy + cy2.In chapter I we introduce the main results.In chapter 2,for the congruence equation f(x,y)? c mod n,when x or y in(Z/nZ)x or not,we give the number of zeros of the equation.Then,we do the same thing to x1x2...xt? c mod n.At the end of this chapter,we study the factorization of f(x,y)mod n and obtain an algorithm for it.Chapter 3 is a research report about the computation of the class numbers of orders in imaginary quadratic fields,following the book of Cohen(A Course in Computational Algebraic Number Theory).The main purpose is to find a quotient set of some special quadratic forms which has the same cardinality as the order's class group.At the end,we also intrduce another two different methods to compute the class numbers.