Representations by a binary quadratic form with class number 4

Abstract

A homogeneous quadratic polynomial F(x,y)=ax^2+bxy+cy^2 is called a binary quadratic form. In this thesis, we consider the binary form F(x,y)=x^2+64y^2 which has class number 4. Our aim is to give an explicit closed formula for the equation F(x,y)=n for any integer n. To do this, we adopt the method developed in S.-Y. Min and B.-K. Oh[3]. In section 5, we collect all results proved in the previous sections and provide a closed formula of the above equation explicitly.